SherbyNet - /page/wikibook/entry.mu

Micron Document

<!DOCTYPE html>

<title>Local linearization method</title>

</head>

Local linearization method

</h1>

</header>

In <a href="Numerical_analysis" title="Numerical analysis">numerical analysis</a>, the local linearization (LL) method is a general strategy for designing <a href="Numerical_integration" title="Numerical integration">numerical integrators</a> for differential equations based on a local (piecewise) linearization of the given equation on consecutive time intervals. The numerical integrators are then iteratively defined as the solution of the resulting piecewise linear equation at the end of each consecutive interval. The LL method has been developed for a variety of equations such as the <a href="Ordinary_differential_equation" title="Ordinary differential equation">ordinary</a>, <a href="Delay_differential_equation" title="Delay differential equation">delayed</a>, random and <a href="Stochastic_differential_equation" title="Stochastic differential equation">stochastic</a> differential equations. The LL integrators are key component in the implementation of <a href="Estimation_theory" title="Estimation theory">inference methods</a> for the estimation of unknown parameters and unobserved variables of differential equations given <a href="Time_series" title="Time series">time series</a> of (potentially noisy) observations. The LL schemes are ideals to deal with complex models in a variety of fields as <a href="Neuroscience" title="Neuroscience">neuroscience</a>, <a href="Finance" title="Finance">finance</a>, <a href="Forestry" title="Forestry">forestry management</a>, <a href="Control_engineering" title="Control engineering">control engineering</a>, <a href="Mathematical_statistics" title="Mathematical statistics">mathematical statistics</a>, etc.

<div class="mw-heading mw-heading2"><h2 id="Background">Background</h2></div>

Differential equations have become an important mathematical tool for describing the time evolution of several phenomenon, e.g., rotation of the planets around the sun, the dynamic of assets prices in the market, the fire of neurons, the propagation of epidemics, etc. However, since the exact solutions of these equations are usually unknown, numerical approximations to them obtained by numerical integrators are necessary. Currently, many applications in engineering and applied sciences focused in dynamical studies demand the developing of efficient numerical integrators that preserve, as much as possible, the dynamics of these equations. With this main motivation, the Local Linearization integrators have been developed.

<div class="mw-heading mw-heading2"><h2 id="High-order_local_linearization_method">High-order local linearization method</h2></div>

High-order local linearization (HOLL) method is a generalization of the Local Linearization method oriented to obtain high-order integrators for differential equations that preserve the <a href="Stability_theory" title="Stability theory">stability</a> and <a href="Dynamical_system" title="Dynamical system">dynamics</a> of the linear equations. The integrators are obtained by splitting, on consecutive time intervals, the solution x of the original equation in two parts: the solution z of the locally linearized equation plus a high-order approximation of the residual <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} =\mathbf {x} -\mathbf {z} }">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} =\mathbf {x} -\mathbf {z} }</annotation>

</semantics>

</math><img src="./5561feed6b78508caec81ea42fc8920fadc3263c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.64ex; height:2.176ex;" alt="{\displaystyle \mathbf {r} =\mathbf {x} -\mathbf {z} }" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Local_linearization_scheme">Local linearization scheme</h2></div>

A Local Linearization (LL) scheme is the final <a href="Recursion_(computer_science)" title="Recursion (computer science)">recursive algorithm</a> that allows the numerical implementation of a <a href="Discretization" title="Discretization">discretization</a> derived from the LL or HOLL method for a class of differential equations.

<div class="mw-heading mw-heading2"><h2 id="LL_methods_for_ODEs">LL methods for ODEs</h2></div>

Consider the d-dimensional <a href="Ordinary_differential_equation" title="Ordinary differential equation">Ordinary Differential Equation</a> (ODE)

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {d\mathbf {x} \left(t\right)}{dt}}=\mathbf {f} \left(t,\mathbf {x} \left(t\right)\right),\qquad t\in \left[t_{0},T\right],\qquad \qquad \qquad \qquad (4.1)}</annotation>

</semantics>

</math><img src="./b689077ae401751f25375eb3338ed66cc57f756a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:59.555ex; height:5.843ex;" alt="{\displaystyle {\frac {d\mathbf {x} \left(t\right)}{dt}}=\mathbf {f} \left(t,\mathbf {x} \left(t\right)\right),\qquad t\in \left[t_{0},T\right],\qquad \qquad \qquad \qquad (4.1)}" loading="lazy">

</div>

with initial condition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} (t_{0})=\mathbf {x} _{0}}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} (t_{0})=\mathbf {x} _{0}}</annotation>

</semantics>

</math><img src="./4e9b78fd114617e57aff79fcd2713d7760fa660d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.678ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} (t_{0})=\mathbf {x} _{0}}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {f} }</annotation>

</semantics>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(t\right)_{h}=\{t_{n}:n=0,..,N\}}">

<msub>

<mrow>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(t\right)_{h}=\{t_{n}:n=0,..,N\}}</annotation>

</semantics>

</math><img src="./11daa16c857a3312423459d5c471f211ea9f519c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.102ex; height:3.009ex;" alt="{\displaystyle \left(t\right)_{h}=\{t_{n}:n=0,..,N\}}" loading="lazy"> be a time discretization of the time interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [t_{0},T]}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [t_{0},T]}</annotation>

</semantics>

</math><img src="./986ba7ea2bc36ce31beb5c5f4faffbfb6405f69b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.858ex; height:2.843ex;" alt="{\displaystyle [t_{0},T]}" loading="lazy"> with maximum stepsize h such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{n}<t_{n+1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{n}<t_{n+1}}</annotation>

</semantics>

</math><img src="./d6db26b179177abb50bf8e3392109b39840dc53a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.315ex; height:2.343ex;" alt="{\displaystyle t_{n}<t_{n+1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{n}=t_{n+1}-t_{n}\leq h}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h_{n}=t_{n+1}-t_{n}\leq h}</annotation>

</semantics>

</math><img src="./b572075cd16fb2d8c938b8b2bacb556028193f47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.15ex; height:2.509ex;" alt="{\displaystyle h_{n}=t_{n+1}-t_{n}\leq h}" loading="lazy">. After the local linearization of the equation (4.1) at the time step <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{n}}</annotation>

</semantics>

</math><img src="./271566db7e8ca8616a4dc3efb6c5982a2d987ee3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.058ex; height:2.343ex;" alt="{\displaystyle t_{n}}" loading="lazy"> the <a href="Variation_of_parameters#First-order_equation" title="Variation of parameters">variation of constants formula</a> yields

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} (t_{n}+h)=\mathbf {x} (t_{n})+\mathbf {\phi } (t_{n},\mathbf {x} (t_{n});h)+\mathbf {r} (t_{n},\mathbf {x} (t_{n});h),}</annotation>

</semantics>

</math><img src="./b1e872c8595fb890930683ae8dca2c21ba1a19d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.755ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} (t_{n}+h)=\mathbf {x} (t_{n})+\mathbf {\phi } (t_{n},\mathbf {x} (t_{n});h)+\mathbf {r} (t_{n},\mathbf {x} (t_{n});h),}" loading="lazy">

</div>

where

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\phi } (t_{n},\mathbf {z} _{n};h)=\int \limits _{0}^{h}e^{\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {z} _{n})(h-s)}(\mathbf {f} (t_{n},\mathbf {z} _{n})+\mathbf {f} _{t}(t_{n},\mathbf {z} _{n})s)\,ds\qquad }</annotation>

</semantics>

</math><img src="./59a781241dc862eb659e9ed49e20bab5300525cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:59.379ex; height:9.343ex;" alt="{\displaystyle \mathbf {\phi } (t_{n},\mathbf {z} _{n};h)=\int \limits _{0}^{h}e^{\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {z} _{n})(h-s)}(\mathbf {f} (t_{n},\mathbf {z} _{n})+\mathbf {f} _{t}(t_{n},\mathbf {z} _{n})s)\,ds\qquad }" loading="lazy">

</div>

results from the linear approximation, and

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} (t_{n},\mathbf {z} _{n};h)=\int \limits _{0}^{h}e^{\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {z} _{n})(h-s)}\mathbf {g} _{n}(s,\mathbf {x} (t_{n}+s))\,ds,\qquad \qquad \qquad (4.2)}</annotation>

</semantics>

</math><img src="./fd1d03a225e9af2b1be2e03b8e3774993d987557.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:66.847ex; height:9.343ex;" alt="{\displaystyle \mathbf {r} (t_{n},\mathbf {z} _{n};h)=\int \limits _{0}^{h}e^{\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {z} _{n})(h-s)}\mathbf {g} _{n}(s,\mathbf {x} (t_{n}+s))\,ds,\qquad \qquad \qquad (4.2)}" loading="lazy">

</div>

is the residual of the linear approximation. Here, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} _{\mathbf {x} }}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {f} _{\mathbf {x} }}</annotation>

</semantics>

</math><img src="./0d1fff5ced74f815fed8a4b4f1f66b154f7e0ccb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.046ex; height:2.509ex;" alt="{\displaystyle \mathbf {f} _{\mathbf {x} }}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} _{t}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {f} _{t}}</annotation>

</semantics>

</math><img src="./bc2e23c23bb2fe729cadf4ef26edef6c768473fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.642ex; height:2.509ex;" alt="{\displaystyle \mathbf {f} _{t}}" loading="lazy"> denote the partial derivatives of f with respect to the variables x and t, respectively, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {g} _{n}(s,\mathbf {u} )=\mathbf {f} (s,\mathbf {u} )-\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {z} _{n})\mathbf {u} -\mathbf {f} _{t}(t_{n},\mathbf {z} _{n})(s-t_{n})-\mathbf {f} (t_{n},\mathbf {z} _{n})+\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {z} _{n})\mathbf {z} _{n}.}">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {g} _{n}(s,\mathbf {u} )=\mathbf {f} (s,\mathbf {u} )-\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {z} _{n})\mathbf {u} -\mathbf {f} _{t}(t_{n},\mathbf {z} _{n})(s-t_{n})-\mathbf {f} (t_{n},\mathbf {z} _{n})+\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {z} _{n})\mathbf {z} _{n}.}</annotation>

</semantics>

</math><img src="./9a48ef9b7048967b82c9a42a352a9bfc6ac78838.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:77.263ex; height:2.843ex;" alt="{\displaystyle \mathbf {g} _{n}(s,\mathbf {u} )=\mathbf {f} (s,\mathbf {u} )-\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {z} _{n})\mathbf {u} -\mathbf {f} _{t}(t_{n},\mathbf {z} _{n})(s-t_{n})-\mathbf {f} (t_{n},\mathbf {z} _{n})+\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {z} _{n})\mathbf {z} _{n}.}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Local_linear_discretization">Local linear discretization</h3></div>

For a time discretization <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(t\right)_{h}}">

<msub>

<mrow>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(t\right)_{h}}</annotation>

</semantics>

</math><img src="./16c122098f93662ae323d6fe099721b11f92825b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.828ex; height:3.009ex;" alt="{\displaystyle \left(t\right)_{h}}" loading="lazy">, the Local Linear discretization of the ODE (4.1) at each point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{n+1}\in \left(t\right)_{h}}">

<msub>

</mrow>

</msub>

<msub>

<mrow>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{n+1}\in \left(t\right)_{h}}</annotation>

</semantics>

</math><img src="./59bd1e8eb52a954e56d94e1236bbcd920f78110d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.827ex; height:3.009ex;" alt="{\displaystyle t_{n+1}\in \left(t\right)_{h}}" loading="lazy"> is defined by the recursive expression <a href="#cite_note-:8-1">[1]</a><a href="#cite_note-:18-2">[2]</a>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} _{n+1}=\mathbf {z} _{n}+\mathbf {\phi } (t_{n},\mathbf {z} _{n};h_{n}),\qquad {\text{ with }}\quad \mathbf {z} _{0}=\mathbf {x} _{0}.\qquad \qquad \qquad \qquad (4.3)}</annotation>

</semantics>

</math><img src="./7af48d951e97f1d600265e5f6f902f0877578c56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:71.024ex; height:2.843ex;" alt="{\displaystyle \mathbf {z} _{n+1}=\mathbf {z} _{n}+\mathbf {\phi } (t_{n},\mathbf {z} _{n};h_{n}),\qquad {\text{ with }}\quad \mathbf {z} _{0}=\mathbf {x} _{0}.\qquad \qquad \qquad \qquad (4.3)}" loading="lazy">

</div>

The Local Linear discretization (4.3) <a href="Rate_of_convergence" title="Rate of convergence">converges</a> with order 2 to the solution of nonlinear ODEs, but it match the solution of the linear ODEs. The recursion (4.3) is also known as Exponential Euler discretization.<a href="#cite_note-:22-3">[3]</a>

<div class="mw-heading mw-heading3"><h3 id="High-order_local_linear_discretizations">High-order local linear discretizations</h3></div>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (t)_{h},}</annotation>

</semantics>

</math><img src="./71717dba289a8ebfd68c0633fd68016d853cfb04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.475ex; height:2.843ex;" alt="{\displaystyle (t)_{h},}" loading="lazy"> a high-order local linear (HOLL) discretization of the ODE (4.1) at each point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{n+1}\in (t)_{h}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{n+1}\in (t)_{h}}</annotation>

</semantics>

</math><img src="./a2c199227f90c5987811cd263767a88826754f50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.827ex; height:2.843ex;" alt="{\displaystyle t_{n+1}\in (t)_{h}}" loading="lazy"> is defined by the recursive expression <a href="#cite_note-:8-1">[1]</a><a href="#cite_note-:3-4">[4]</a><a href="#cite_note-:2-5">[5]</a><a href="#cite_note-:1-6">[6]</a>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} _{n+1}=\mathbf {z} _{n}+\mathbf {\phi } (t_{n},\mathbf {z} _{n};h_{n})+{\widetilde {\mathbf {r} }}(t_{n},\mathbf {z} _{n};h_{n}),\qquad {\text{ with }}\quad \mathbf {z} _{0}=\mathbf {x} _{0},\qquad \qquad \qquad (4.4)}</annotation>

</semantics>

</math><img src="./5f7165c136dba0e51ad29ba229d4eb71feac245b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:81.281ex; height:2.843ex;" alt="{\displaystyle \mathbf {z} _{n+1}=\mathbf {z} _{n}+\mathbf {\phi } (t_{n},\mathbf {z} _{n};h_{n})+{\widetilde {\mathbf {r} }}(t_{n},\mathbf {z} _{n};h_{n}),\qquad {\text{ with }}\quad \mathbf {z} _{0}=\mathbf {x} _{0},\qquad \qquad \qquad (4.4)}" loading="lazy">

</div>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {r}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {r}}}</annotation>

</semantics>

</math><img src="./85962e2c63f61329fa0089221292e8e519c9fe2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.292ex; height:2.176ex;" alt="{\displaystyle {\tilde {r}}}" loading="lazy"> is an order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> (> 2) approximation to the residual r <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i.e.,\left\vert \mathbf {r} (t_{n},\mathbf {z} _{n};h)-{\widetilde {\mathbf {r} }}(t_{n},\mathbf {z} _{n};h)\right\vert \propto h^{\alpha +1}).}">

<mrow>

<mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (i.e.,\left\vert \mathbf {r} (t_{n},\mathbf {z} _{n};h)-{\widetilde {\mathbf {r} }}(t_{n},\mathbf {z} _{n};h)\right\vert \propto h^{\alpha +1}).}</annotation>

</semantics>

</math><img src="./fc801c2f0b037489ed6257b76784695a70325003.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.026ex; height:3.176ex;" alt="{\displaystyle (i.e.,\left\vert \mathbf {r} (t_{n},\mathbf {z} _{n};h)-{\widetilde {\mathbf {r} }}(t_{n},\mathbf {z} _{n};h)\right\vert \propto h^{\alpha +1}).}" loading="lazy"> The HOLL discretization (4.4) <a href="Rate_of_convergence" title="Rate of convergence">converges</a> with order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

HOLL discretizations can be derived in two ways:<a href="#cite_note-:8-1">[1]</a><a href="#cite_note-:3-4">[4]</a><a href="#cite_note-:2-5">[5]</a><a href="#cite_note-:1-6">[6]</a> 1) (quadrature-based) by approximating the integral representation (4.2) of r; and 2) (integrator-based) by using a numerical integrator for the differential representation of r defined by

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {d\mathbf {r} (t)}{dt}}=\mathbf {q} (t_{n},\mathbf {z} _{n};t\mathbf {,\mathbf {r} } (t)\mathbf {)} ,\qquad {\text{ with }}\qquad \mathbf {r} (t_{n})=\mathbf {0,} \qquad \qquad \qquad (4.5)}</annotation>

</semantics>

</math><img src="./12166458f1c6b042b1009eb033641773708f37cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:69.384ex; height:5.843ex;" alt="{\displaystyle {\frac {d\mathbf {r} (t)}{dt}}=\mathbf {q} (t_{n},\mathbf {z} _{n};t\mathbf {,\mathbf {r} } (t)\mathbf {)} ,\qquad {\text{ with }}\qquad \mathbf {r} (t_{n})=\mathbf {0,} \qquad \qquad \qquad (4.5)}" loading="lazy">

</div>

for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in \lbrack t_{k},t_{k+1}]}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t\in \lbrack t_{k},t_{k+1}]}</annotation>

</semantics>

</math><img src="./dacd0fb31482f9ac95b8c3261d9dc00f1055dca4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.965ex; height:2.843ex;" alt="{\displaystyle t\in \lbrack t_{k},t_{k+1}]}" loading="lazy">, where

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {q} (t_{n},\mathbf {z} _{n};s\mathbf {,\xi } )=\mathbf {f} (s,\mathbf {z} _{n}+\mathbf {\phi } \left(t_{n},\mathbf {z} _{n};s-t_{n}\right)+\mathbf {\xi } )-\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {z} _{n})\mathbf {\phi } (t_{n},\mathbf {z} _{n};s-t_{n})-\mathbf {f} _{t}(t_{n},\mathbf {z} _{n})(s-t_{n})-\mathbf {f} (t_{n},\mathbf {z} _{n}).}</annotation>

</semantics>

</math><img src="./a4f669a8ca1005f8d5635c2dccbb9b9e2f423cf5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:105.661ex; height:2.843ex;" alt="{\displaystyle \mathbf {q} (t_{n},\mathbf {z} _{n};s\mathbf {,\xi } )=\mathbf {f} (s,\mathbf {z} _{n}+\mathbf {\phi } \left(t_{n},\mathbf {z} _{n};s-t_{n}\right)+\mathbf {\xi } )-\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {z} _{n})\mathbf {\phi } (t_{n},\mathbf {z} _{n};s-t_{n})-\mathbf {f} _{t}(t_{n},\mathbf {z} _{n})(s-t_{n})-\mathbf {f} (t_{n},\mathbf {z} _{n}).}" loading="lazy">

</div>

HOLL discretizations are, for instance, the followings:

<ul><li>Locally Linearized Runge Kutta discretization<a href="#cite_note-:1-6">[6]</a><a href="#cite_note-:3-4">[4]</a></li></ul>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \qquad \mathbf {z} _{n+1}=\mathbf {z} _{n}+\mathbf {\phi } (t_{n},\mathbf {z} _{n};h_{n})+h_{n}\sum _{j=1}^{s}b_{j}\mathbf {k} _{j},\quad {\text{ with }}\quad \mathbf {k} _{i}=\mathbf {q} (t_{n},\mathbf {z} _{n};{\text{ }}t_{n}+c_{i}h_{n}\mathbf {,} \mathbf {} h_{n}\sum _{j=1}^{i-1}a_{ij}\mathbf {k} _{j}),}</annotation>

</semantics>

</math><img src="./b1c5f0fe2adaa2ffbab05b071aa59f4833395c59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:92.912ex; height:7.676ex;" alt="{\displaystyle \qquad \mathbf {z} _{n+1}=\mathbf {z} _{n}+\mathbf {\phi } (t_{n},\mathbf {z} _{n};h_{n})+h_{n}\sum _{j=1}^{s}b_{j}\mathbf {k} _{j},\quad {\text{ with }}\quad \mathbf {k} _{i}=\mathbf {q} (t_{n},\mathbf {z} _{n};{\text{ }}t_{n}+c_{i}h_{n}\mathbf {,} \mathbf {} h_{n}\sum _{j=1}^{i-1}a_{ij}\mathbf {k} _{j}),}" loading="lazy">

</div>

which is obtained by solving (4.5) via a s-stage explicit <a href="Runge%E2%80%93Kutta_methods" title="Runge–Kutta methods">Runge–Kutta (RK) scheme</a> with coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {c} =\left[c_{i}\right],\mathbf {A} =\left[a_{ij}\right]\quad and\quad \mathbf {b} =\left[b_{j}\right]}">

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {c} =\left[c_{i}\right],\mathbf {A} =\left[a_{ij}\right]\quad and\quad \mathbf {b} =\left[b_{j}\right]}</annotation>

</semantics>

</math><img src="./6fbc317004aaf306e62ecc1c6d08e4f49d9d0543.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.584ex; height:3.009ex;" alt="{\displaystyle \mathbf {c} =\left[c_{i}\right],\mathbf {A} =\left[a_{ij}\right]\quad and\quad \mathbf {b} =\left[b_{j}\right]}" loading="lazy">.

<ul><li>Local linear Taylor discretization<a href="#cite_note-:2-5">[5]</a></li></ul>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

<msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<mfrac>

<msub>

</mrow>

</mrow>

</msub>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

<msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} _{n+1}=\mathbf {z} _{n}+\mathbf {\phi } (t_{n},\mathbf {z} _{n};h_{n})+\int _{0}^{h_{n}}e^{(h_{n}-s)\mathbf {f} _{\mathbf {x} }\left(t_{n},\mathbf {z} _{n}\right)}\sum _{j=2}^{p}{\frac {\mathbf {c} _{n,j}}{j!}}s^{j}\,ds,{\text{ with }}\mathbf {c} _{n,j}=\left({\frac {d^{j+1}\mathbf {x} (t)}{dt^{j+1}}}-\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {z} _{n}){\frac {d^{j}\mathbf {x} (t)}{dt^{j}}}\right)\mid _{t=\mathbf {z} _{n}},}</annotation>

</semantics>

</math>

</div>

which results from the approximation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {g} _{n}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {g} _{n}}</annotation>

</semantics>

<ul><li>Multistep-type exponential propagation discretization</li></ul>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</msup>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mathbf {z} _{n+1}=\mathbf {z} _{n}+\mathbf {\phi } (t_{n},\mathbf {z} _{n};h)+h\sum _{j=0}^{p-1}\gamma _{j}\nabla ^{j}\mathbf {g} _{n}(t_{n},\mathbf {z} _{n}),\quad with\quad \gamma _{j}=(-1)^{j}\int \limits _{0}^{1}e^{(1-\theta )h\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {z} _{n})}\left({\begin{array}{c}-\theta \\j\end{array}}\right)d\theta ,}</annotation>

</semantics>

</math><img src="./79193a9a9f809161b97cf41b3a983e71b5d21c31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:96.59ex; height:6.509ex;" alt="{\textstyle \mathbf {z} _{n+1}=\mathbf {z} _{n}+\mathbf {\phi } (t_{n},\mathbf {z} _{n};h)+h\sum _{j=0}^{p-1}\gamma _{j}\nabla ^{j}\mathbf {g} _{n}(t_{n},\mathbf {z} _{n}),\quad with\quad \gamma _{j}=(-1)^{j}\int \limits _{0}^{1}e^{(1-\theta )h\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {z} _{n})}\left({\begin{array}{c}-\theta \\j\end{array}}\right)d\theta ,}" loading="lazy">

</div>

which results from the interpolation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {g} _{n}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {g} _{n}}</annotation>

</semantics>

</math><img src="./deaee9834d3c5ccba2220faa77fb05add408c92a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.555ex; height:2.176ex;" alt="{\displaystyle \mathbf {g} _{n}}" loading="lazy"> in (4.2) by a polynomial of degree p on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{n},\ldots ,t_{n-p+1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{n},\ldots ,t_{n-p+1}}</annotation>

</semantics>

</math><img src="./1a877dc54fc46a1e1e6f996a99fe2197c013f738.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.5ex; height:2.676ex;" alt="{\displaystyle t_{n},\ldots ,t_{n-p+1}}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla ^{j}\mathbf {g} _{n}(t_{m},\mathbf {z} _{m})}">

<msup>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nabla ^{j}\mathbf {g} _{n}(t_{m},\mathbf {z} _{m})}</annotation>

</semantics>

</math><img src="./3f29a213b85c448022398faddfbd0bcd09dd524c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.622ex; height:3.176ex;" alt="{\displaystyle \nabla ^{j}\mathbf {g} _{n}(t_{m},\mathbf {z} _{m})}" loading="lazy"> denotes the j-th <a href="Backward_difference" class="mw-redirect" title="Backward difference">backward difference</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {g} _{n}(t_{m},\mathbf {z} _{m})}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {g} _{n}(t_{m},\mathbf {z} _{m})}</annotation>

</semantics>

</math><img src="./78f3249a37f4ac5b0f5f381c23048f046db32be1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.776ex; height:2.843ex;" alt="{\displaystyle \mathbf {g} _{n}(t_{m},\mathbf {z} _{m})}" loading="lazy">.

<ul><li>Runge Kutta type Exponential Propagation discretization <a href="#cite_note-:17-7">[7]</a></li></ul>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</msup>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mathbf {z} _{n+1}=\mathbf {z} _{n}+\mathbf {\phi } (t_{n},\mathbf {z} _{n};h)+h\sum _{j=0}^{p-1}\gamma _{j,p}\nabla ^{j}\mathbf {g} _{n}(t_{n},\mathbf {z} _{n}),\quad {\text{ with }}\quad \gamma _{j,p}=\int \limits _{0}^{1}e^{(1-\theta )h\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {z} _{n})}\left({\begin{array}{c}\theta p\\j\end{array}}\right)d\theta ,}</annotation>

</semantics>

</math><img src="./2d4c8de3f94e78b8c970fcf15585ce1007e22577.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:93.482ex; height:6.509ex;" alt="{\textstyle \mathbf {z} _{n+1}=\mathbf {z} _{n}+\mathbf {\phi } (t_{n},\mathbf {z} _{n};h)+h\sum _{j=0}^{p-1}\gamma _{j,p}\nabla ^{j}\mathbf {g} _{n}(t_{n},\mathbf {z} _{n}),\quad {\text{ with }}\quad \gamma _{j,p}=\int \limits _{0}^{1}e^{(1-\theta )h\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {z} _{n})}\left({\begin{array}{c}\theta p\\j\end{array}}\right)d\theta ,}" loading="lazy">

</div>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {g} _{n}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{n},\ldots ,t_{n}+(p-1)h/p}</annotation>

</semantics>

</math><img src="./7c31429448e76501407e0049dc50ced708d17d2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.787ex; height:2.843ex;" alt="{\displaystyle t_{n},\ldots ,t_{n}+(p-1)h/p}" loading="lazy">,

<ul><li>Linealized exponential Adams discretization<a href="#cite_note-:7-8">[8]</a></li></ul>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</munderover>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msup>

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mathbf {z} _{n+1}=\mathbf {z} _{n}+\mathbf {\phi } (t_{n},\mathbf {z} _{n};h)+h\sum _{j=1}^{p-1}\sum _{l=1}^{j}{\frac {\gamma _{j+1}}{l}}\nabla ^{l}\mathbf {g} _{n}(t_{n},\mathbf {z} _{n}),\quad {\text{ with }}\quad \gamma _{j+1}=(-1)^{j+1}\int \limits _{0}^{1}e^{(1-\theta )h\mathbf {f} _{\mathbf {x} }\left(t_{n},\mathbf {z} _{n}\right)}\theta \left({\begin{array}{c}-\theta \\j\end{array}}\right)d\theta ,}</annotation>

</semantics>

</math><img src="./54d62daf1e988553798545a5deaa44652a6ccae7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:110.39ex; height:6.509ex;" alt="{\textstyle \mathbf {z} _{n+1}=\mathbf {z} _{n}+\mathbf {\phi } (t_{n},\mathbf {z} _{n};h)+h\sum _{j=1}^{p-1}\sum _{l=1}^{j}{\frac {\gamma _{j+1}}{l}}\nabla ^{l}\mathbf {g} _{n}(t_{n},\mathbf {z} _{n}),\quad {\text{ with }}\quad \gamma _{j+1}=(-1)^{j+1}\int \limits _{0}^{1}e^{(1-\theta )h\mathbf {f} _{\mathbf {x} }\left(t_{n},\mathbf {z} _{n}\right)}\theta \left({\begin{array}{c}-\theta \\j\end{array}}\right)d\theta ,}" loading="lazy">

</div>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {g} _{n}}</annotation>

</semantics>

</math><img src="./deaee9834d3c5ccba2220faa77fb05add408c92a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.555ex; height:2.176ex;" alt="{\displaystyle \mathbf {g} _{n}}" loading="lazy"> in (4.2) by a <a href="Hermite_polynomials" title="Hermite polynomials">Hermite polynomial</a> of degree p on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{n},\ldots ,t_{n-p+1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{n},\ldots ,t_{n-p+1}}</annotation>

</semantics>

<div class="mw-heading mw-heading3"><h3 id="Local_linearization_schemes">Local linearization schemes</h3></div>

All numerical implementation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} _{n}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{n}}</annotation>

</semantics>

</math><img src="./05275acac23ef658db9a5d0b777828444fa2096c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.629ex; height:2.176ex;" alt="{\displaystyle \mathbf {y} _{n}}" loading="lazy"> of the LL (or of a HOLL) discretization <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {z} _{n}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} _{n}}</annotation>

</semantics>

</math><img src="./ff3c4e54a2f18ad992a5eb8057055d92e1389f5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.406ex; height:2.009ex;" alt="{\displaystyle \mathbf {z} _{n}}" loading="lazy"> involves approximations <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {\phi }}_{j}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widetilde {\phi }}_{j}}</annotation>

</semantics>

</math><img src="./2173a6bd2483a87e8eeca7925fb2b7e410b42865.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.442ex; height:3.509ex;" alt="{\displaystyle {\widetilde {\phi }}_{j}}" loading="lazy"> to integrals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi _{j}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi _{j}(\mathbf {A} ,h)=\int \limits _{0}^{h}e^{(h-s)\mathbf {A} }s^{j-1}\,ds,\qquad j=1,2\ldots ,}</annotation>

</semantics>

</math><img src="./db7430144b6412098ed3d547bc65d9e2ce735ce5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:45.229ex; height:9.343ex;" alt="{\displaystyle \phi _{j}(\mathbf {A} ,h)=\int \limits _{0}^{h}e^{(h-s)\mathbf {A} }s^{j-1}\,ds,\qquad j=1,2\ldots ,}" loading="lazy">

</div>

where A is a d × d matrix. Every numerical implementation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} _{n}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{n}}</annotation>

</semantics>

</math><img src="./05275acac23ef658db9a5d0b777828444fa2096c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.629ex; height:2.176ex;" alt="{\displaystyle \mathbf {y} _{n}}" loading="lazy"> of the LL (or of a HOLL) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {z} _{n}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} _{n}}</annotation>

</semantics>

</math><img src="./ff3c4e54a2f18ad992a5eb8057055d92e1389f5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.406ex; height:2.009ex;" alt="{\displaystyle \mathbf {z} _{n}}" loading="lazy"> of any order is generically called Local Linearization scheme.<a href="#cite_note-:8-1">[1]</a><a href="#cite_note-:25-9">[9]</a>

<div class="mw-heading mw-heading4"><h4 id="Computing_integrals_involving_matrix_exponential">Computing integrals involving matrix exponential</h4></div>

Among a number of algorithms to compute the integrals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi _{j}}</annotation>

</semantics>

</math><img src="./e4cbf9f77e99bf49bdc5b3afa8b6b77d15b9be6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.295ex; height:2.843ex;" alt="{\displaystyle \phi _{j}}" loading="lazy">, those based on rational Padé and Krylov subspaces approximations for exponential matrix are preferred. For this, a central role is playing by the expression<a href="#cite_note-10">[10]</a><a href="#cite_note-:2-5">[5]</a><a href="#cite_note-:13-11">[11]</a>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum \nolimits _{i=1}^{l}\phi _{i}(\mathbf {A} ,h)\mathbf {a} _{i}=\mathbf {L} e^{h\mathbf {H} }\mathbf {r} ,}</annotation>

</semantics>

</math><img src="./fc389451a8c0eecbf86cfe7bc04b5c6bb798c710.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:27.325ex; height:4.343ex;" alt="{\displaystyle \sum \nolimits _{i=1}^{l}\phi _{i}(\mathbf {A} ,h)\mathbf {a} _{i}=\mathbf {L} e^{h\mathbf {H} }\mathbf {r} ,}" loading="lazy">

</div>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} _{i}}</annotation>

</semantics>

</math><img src="./0a01879ce830ef8790aa7dc9f3665d6727f3af3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.099ex; height:2.009ex;" alt="{\displaystyle \mathbf {a} _{i}}" loading="lazy"> are d-dimensional vectors,

</mrow>

<mrow>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {H} ={\begin{bmatrix}\mathbf {A} &\mathbf {v} _{l}&\mathbf {v} _{l-1}&\cdots &\mathbf {v} _{1}\\\mathbf {0} &\mathbf {0} &1&\cdots &0\\\mathbf {0} &\mathbf {0} &0&\ddots &0\\\vdots &\vdots &\vdots &\ddots &1\\\mathbf {0} &\mathbf {0} &0&\cdots &0\end{bmatrix}}\in \mathbb {R} ^{(d+l)\times (d+l)},}</annotation>

</semantics>

</math><img src="./c7ffd88d87d61eba3ae08960aca3fa505c95d331.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:46.655ex; height:19.176ex;" alt="{\displaystyle \mathbf {H} ={\begin{bmatrix}\mathbf {A} &\mathbf {v} _{l}&\mathbf {v} _{l-1}&\cdots &\mathbf {v} _{1}\\\mathbf {0} &\mathbf {0} &1&\cdots &0\\\mathbf {0} &\mathbf {0} &0&\ddots &0\\\vdots &\vdots &\vdots &\ddots &1\\\mathbf {0} &\mathbf {0} &0&\cdots &0\end{bmatrix}}\in \mathbb {R} ^{(d+l)\times (d+l)},}" loading="lazy">

</div>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {L} =[\mathbf {I} \quad \mathbf {0} _{d\times l}]}</annotation>

</semantics>

</math><img src="./75b8ac2636b6ebd86e0bba8f06886336cfc6ec82.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.534ex; height:2.843ex;" alt="{\displaystyle \mathbf {L} =[\mathbf {I} \quad \mathbf {0} _{d\times l}]}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} =[\mathbf {0} _{1\times (d+l-1)}\quad 1]^{\intercal },}">

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} =[\mathbf {0} _{1\times (d+l-1)}\quad 1]^{\intercal },}</annotation>

</semantics>

</math><img src="./3710227203d25399ba90fb8e8f94a9d34ff29aa9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.45ex; height:3.176ex;" alt="{\displaystyle \mathbf {r} =[\mathbf {0} _{1\times (d+l-1)}\quad 1]^{\intercal },}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{i}=\mathbf {a} _{i}(i-1)!}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{i}=\mathbf {a} _{i}(i-1)!}</annotation>

</semantics>

</math><img src="./a5495f8e3af608f2f1cf855ca3b39bafabea7c20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.67ex; height:2.843ex;" alt="{\displaystyle \mathbf {v} _{i}=\mathbf {a} _{i}(i-1)!}" loading="lazy">, being <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {I} }</annotation>

</semantics>

</math><img src="./8a458c8aeb096ce732abf346ae8edf3e4f53a126.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.014ex; height:2.176ex;" alt="{\displaystyle \mathbf {I} }" loading="lazy"> the d-dimensional identity matrix.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} _{p,q}(2^{-k}\mathbf {H} h)}">

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} _{p,q}(2^{-k}\mathbf {H} h)}</annotation>

</semantics>

</math><img src="./b6e308c626ec6e3f96de2df6631e18567b82466e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.869ex; height:3.343ex;" alt="{\displaystyle \mathbf {P} _{p,q}(2^{-k}\mathbf {H} h)}" loading="lazy"> denotes the (p; q)-<a href="Pad%C3%A9_approximant" title="Padé approximant">Padé approximation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{2^{-k}\mathbf {H} h}}">

<msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{2^{-k}\mathbf {H} h}}</annotation>

</semantics>

</math><img src="./055edc7b8e41df6f816fbd81fecab6632b7f0e7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.461ex; height:3.009ex;" alt="{\displaystyle e^{2^{-k}\mathbf {H} h}}" loading="lazy"> and k is the smallest natural number such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |2^{-k}\mathbf {H} h|\leq {\frac {1}{2}},then}">

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |2^{-k}\mathbf {H} h|\leq {\frac {1}{2}},then}</annotation>

</semantics>

</math><img src="./563d579272619dc379c9985f3cd93c13cf0a2702.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.042ex; height:5.176ex;" alt="{\displaystyle |2^{-k}\mathbf {H} h|\leq {\frac {1}{2}},then}" loading="lazy"> <a href="#cite_note-:12-12">[12]</a><a href="#cite_note-:25-9">[9]</a>

<mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msup>

<mrow>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\vert \sum \nolimits _{i=1}^{l}\phi _{i}(\mathbf {A} ,h)\mathbf {a} _{i}-\mathbf {L} \left(\mathbf {\mathbf {P} } _{p,q}(2^{-k}\mathbf {H} h)\right)^{2^{k}}\mathbf {r} \right\vert \varpropto h^{p+q+1}.}</annotation>

</semantics>

</math><img src="./8dc7eb0c41e7cdf2494ca768d14862edccf43996.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:51.164ex; height:5.176ex;" alt="{\displaystyle \left\vert \sum \nolimits _{i=1}^{l}\phi _{i}(\mathbf {A} ,h)\mathbf {a} _{i}-\mathbf {L} \left(\mathbf {\mathbf {P} } _{p,q}(2^{-k}\mathbf {H} h)\right)^{2^{k}}\mathbf {r} \right\vert \varpropto h^{p+q+1}.}" loading="lazy">

</div>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\mathbf {k} } _{m,k}^{p,q}(h,\mathbf {H} ,\mathbf {r} )}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\mathbf {k} } _{m,k}^{p,q}(h,\mathbf {H} ,\mathbf {r} )}</annotation>

</semantics>

</math><img src="./6caa186a3b29f349e006375bef404327bee21570.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.81ex; height:3.509ex;" alt="{\displaystyle \mathbf {\mathbf {k} } _{m,k}^{p,q}(h,\mathbf {H} ,\mathbf {r} )}" loading="lazy"> denotes the (m; p; q; k) <a href="Krylov_subspace" title="Krylov subspace">Krylov-Padé approximation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{h\mathbf {H} }\mathbf {r} }">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{h\mathbf {H} }\mathbf {r} }</annotation>

</semantics>

</math><img src="./15bbe2aa8d2b287389cb1eb8fa2174af2fc5ca92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.844ex; height:2.676ex;" alt="{\displaystyle e^{h\mathbf {H} }\mathbf {r} }" loading="lazy">, then <a href="#cite_note-:12-12">[12]</a>

<mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\vert \sum \nolimits _{i=1}^{l}\phi _{i}(\mathbf {A} ,h)\mathbf {a} _{i}-\mathbf {L\mathbf {k} } _{m,k}^{p,q}(h,\mathbf {H} ,\mathbf {r} )\right\vert \varpropto h^{\min({m,p+q+1})},}</annotation>

</semantics>

</math><img src="./dc35c82a918dba68d9657d979eff959d83390cd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:51.879ex; height:4.843ex;" alt="{\displaystyle \left\vert \sum \nolimits _{i=1}^{l}\phi _{i}(\mathbf {A} ,h)\mathbf {a} _{i}-\mathbf {L\mathbf {k} } _{m,k}^{p,q}(h,\mathbf {H} ,\mathbf {r} )\right\vert \varpropto h^{\min({m,p+q+1})},}" loading="lazy">

</div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m\leq d}</annotation>

</semantics>

</math><img src="./fa506ab9e92a2146e659f438384d6b0f94015264.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.355ex; height:2.343ex;" alt="{\displaystyle m\leq d}" loading="lazy"> is the dimension of the Krylov subspace.

<div class="mw-heading mw-heading4"><h4 id="Order-2_LL_schemes">Order-2 LL schemes</h4></div>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{n+1}=\mathbf {y} _{n}+\mathbf {L} (\mathbf {P} _{p,q}(2^{-k_{n}}\mathbf {M} _{n}h_{n}))^{2^{k_{n}}}\mathbf {r,} }</annotation>

</semantics>

</math><img src="./e16e8214f3772879c0fe96e190a3aa6421557119.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:38.125ex; height:3.676ex;" alt="{\displaystyle \mathbf {y} _{n+1}=\mathbf {y} _{n}+\mathbf {L} (\mathbf {P} _{p,q}(2^{-k_{n}}\mathbf {M} _{n}h_{n}))^{2^{k_{n}}}\mathbf {r,} }" loading="lazy"> <a href="#cite_note-:9-13">[13]</a><a href="#cite_note-:25-9">[9]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad \qquad (4.6)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \qquad \qquad (4.6)}</annotation>

</semantics>

</math><img src="./e3cd268640da6ed147af7a85a6c51a47d10304dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.071ex; height:2.843ex;" alt="{\displaystyle \qquad \qquad (4.6)}" loading="lazy">

</div>

where the matrices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} _{n}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {M} _{n}}</annotation>

</semantics>

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {M} _{n}={\begin{bmatrix}\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {y} _{n})&\mathbf {f} _{t}(t_{n},\mathbf {y} _{n})&\mathbf {f} (t_{n},\mathbf {y} _{n})\\0&0&1\\0&0&0\end{bmatrix}}\in \mathbb {R} ^{(d+2)\times (d+2)},}</annotation>

</semantics>

</math><img src="./8b1e7909afb919dd9b2ce657ba5686be5f529741.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:57.842ex; height:9.509ex;" alt="{\displaystyle \mathbf {M} _{n}={\begin{bmatrix}\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {y} _{n})&\mathbf {f} _{t}(t_{n},\mathbf {y} _{n})&\mathbf {f} (t_{n},\mathbf {y} _{n})\\0&0&1\\0&0&0\end{bmatrix}}\in \mathbb {R} ^{(d+2)\times (d+2)},}" loading="lazy">

</div>

</mrow>

<mrow>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {L} =\left[{\begin{array}{ll}\mathbf {I} &\mathbf {0} _{d\times 2}\end{array}}\right]}</annotation>

</semantics>

</math><img src="./05d54f18477ecdb4b717abf25a75b96d17729bcd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.618ex; height:2.843ex;" alt="{\displaystyle \mathbf {L} =\left[{\begin{array}{ll}\mathbf {I} &\mathbf {0} _{d\times 2}\end{array}}\right]}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} ^{\intercal }=\left[{\begin{array}{ll}\mathbf {0} _{1\times (d+1)}&1\end{array}}\right]}">

<msup>

</mrow>

</mrow>

</msup>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} ^{\intercal }=\left[{\begin{array}{ll}\mathbf {0} _{1\times (d+1)}&1\end{array}}\right]}</annotation>

</semantics>

</math><img src="./f82230fb66e66705351377690e9f14c8d4a41224.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.432ex; height:3.176ex;" alt="{\displaystyle \mathbf {r} ^{\intercal }=\left[{\begin{array}{ll}\mathbf {0} _{1\times (d+1)}&1\end{array}}\right]}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p+q>1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p+q>1}</annotation>

</semantics>

</math><img src="./5340225ba272f9ca32d1fc02c246d11737fed65c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.43ex; height:2.509ex;" alt="{\displaystyle p+q>1}" loading="lazy"> . For large systems of ODEs <a href="#cite_note-:22-3">[3]</a>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

</semantics>

</math><img src="./3b0386903d07d0c90ddba9cf17b504cce91d328c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:56.411ex; height:3.509ex;" alt="{\displaystyle \mathbf {y} _{n+1}=\mathbf {y} _{n}+\mathbf {L\mathbf {k} } _{m_{n},k_{n}}^{p,q}(h_{n},\mathbf {M} _{n},\mathbf {r} )\mathbf {,} \qquad {\text{ with }}\qquad m_{n}>2.}" loading="lazy">

</div>

<div class="mw-heading mw-heading4"><h4 id="Order-3_LL-Taylor_schemes">Order-3 LL-Taylor schemes</h4></div>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{n+1}=\mathbf {y} _{n}+\mathbf {L} _{1}(\mathbf {P} _{p,q}(2^{-k_{n}}\mathbf {T} _{n}h_{n}))^{2^{k_{n}}}\mathbf {r} _{1}\mathbf {,} }</annotation>

</semantics>

</math><img src="./5d7e1e6d67b8670645a893b2bac8eea16ae2b573.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.555ex; height:3.676ex;" alt="{\displaystyle \mathbf {y} _{n+1}=\mathbf {y} _{n}+\mathbf {L} _{1}(\mathbf {P} _{p,q}(2^{-k_{n}}\mathbf {T} _{n}h_{n}))^{2^{k_{n}}}\mathbf {r} _{1}\mathbf {,} }" loading="lazy"> <a href="#cite_note-:2-5">[5]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad \qquad (4.7)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \qquad \qquad (4.7)}</annotation>

</semantics>

</math><img src="./468efe29d89a516e7adb3ae7d0fbefa92b66ebda.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.071ex; height:2.843ex;" alt="{\displaystyle \qquad \qquad (4.7)}" loading="lazy">

</div>

where for <a href="Autonomous_system_(mathematics)" title="Autonomous system (mathematics)">autonomous</a> ODEs the matrices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {T} _{n},\mathbf {L} _{1}}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {T} _{n},\mathbf {L} _{1}}</annotation>

</semantics>

</math><img src="./e4de9e3b10248204ba1ddb81fc5cba510d3f5cf2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.774ex; height:2.509ex;" alt="{\displaystyle \mathbf {T} _{n},\mathbf {L} _{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} _{1}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} _{1}}</annotation>

</semantics>

</math><img src="./c197b71f97673f419ea1a520594f9f7b7c2fe0ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.156ex; height:2.009ex;" alt="{\displaystyle \mathbf {r} _{1}}" loading="lazy"> are defined as

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {T} _{n}=\left[{\begin{array}{cccc}\mathbf {f} _{\mathbf {x} }(\mathbf {y} _{n})&(\mathbf {I} \otimes \mathbf {f} ^{\intercal }(\mathbf {y} _{n}))\mathbf {f} _{\mathbf {xx} }(\mathbf {y} _{n})\mathbf {f} (\mathbf {y} _{n})&\mathbf {0} &\mathbf {f} (\mathbf {y} _{n})\\0&0&0&0\\0&0&0&1\\0&0&0&0\end{array}}\right]\in \mathbb {R} ^{(d+3)\times (d+3)},}</annotation>

</semantics>

</math><img src="./04b4535597fcece8be4c40e7b4ae75229d8f9831.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:70.813ex; height:12.843ex;" alt="{\displaystyle \mathbf {T} _{n}=\left[{\begin{array}{cccc}\mathbf {f} _{\mathbf {x} }(\mathbf {y} _{n})&(\mathbf {I} \otimes \mathbf {f} ^{\intercal }(\mathbf {y} _{n}))\mathbf {f} _{\mathbf {xx} }(\mathbf {y} _{n})\mathbf {f} (\mathbf {y} _{n})&\mathbf {0} &\mathbf {f} (\mathbf {y} _{n})\\0&0&0&0\\0&0&0&1\\0&0&0&0\end{array}}\right]\in \mathbb {R} ^{(d+3)\times (d+3)},}" loading="lazy">

</div>

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {L} _{1}=\left[{\begin{array}{ll}\mathbf {I} &\mathbf {0} _{d\times 3}\end{array}}\right]\quad and\quad \mathbf {r} _{1}^{\intercal }=\left[{\begin{array}{ll}\mathbf {0} _{1\times (d+2)}&1\end{array}}\right]}</annotation>

</semantics>

</math><img src="./f495cb0102ab17316c499936bd236cdb35f4ebbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.976ex; height:3.176ex;" alt="{\displaystyle \mathbf {L} _{1}=\left[{\begin{array}{ll}\mathbf {I} &\mathbf {0} _{d\times 3}\end{array}}\right]\quad and\quad \mathbf {r} _{1}^{\intercal }=\left[{\begin{array}{ll}\mathbf {0} _{1\times (d+2)}&1\end{array}}\right]}" loading="lazy">. Here, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} _{\mathbf {xx} }}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {f} _{\mathbf {xx} }}</annotation>

</semantics>

</math><img src="./0eaeec93fe4af91ad7b0eed791c85ecfa3c87c71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.044ex; height:2.509ex;" alt="{\displaystyle \mathbf {f} _{\mathbf {xx} }}" loading="lazy"> denotes the second derivative of f with respect to x, and p + q > 2. For large systems of ODEs

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{n+1}=\mathbf {y} _{n}+\mathbf {L\mathbf {k} } _{m_{n},k_{n}}^{p,q}(h_{n},\mathbf {T} _{n},\mathbf {r} )\mathbf {,} \qquad {\text{ with }}\qquad m_{n}>3.}</annotation>

</semantics>

</math><img src="./825772609c09d27fe5e738fe1a3b5fd80fb252e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:55.733ex; height:3.509ex;" alt="{\displaystyle \mathbf {y} _{n+1}=\mathbf {y} _{n}+\mathbf {L\mathbf {k} } _{m_{n},k_{n}}^{p,q}(h_{n},\mathbf {T} _{n},\mathbf {r} )\mathbf {,} \qquad {\text{ with }}\qquad m_{n}>3.}" loading="lazy">

</div>

<div class="mw-heading mw-heading4"><h4 id="Order-4_LL-RK_schemes">Order-4 LL-RK schemes</h4></div>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{n+1}=\mathbf {y} _{n}+\mathbf {u} _{4}+{\frac {h_{n}}{6}}(2\mathbf {k} _{2}+2\mathbf {k} _{3}+\mathbf {k} _{4}),\quad }</annotation>

</semantics>

</math><img src="./3afd80fc5546ec15305fc540f92b39f964009961.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:42.252ex; height:5.343ex;" alt="{\displaystyle \mathbf {y} _{n+1}=\mathbf {y} _{n}+\mathbf {u} _{4}+{\frac {h_{n}}{6}}(2\mathbf {k} _{2}+2\mathbf {k} _{3}+\mathbf {k} _{4}),\quad }" loading="lazy"> <a href="#cite_note-:3-4">[4]</a><a href="#cite_note-:1-6">[6]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad \qquad (4.8)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \qquad \qquad (4.8)}</annotation>

</semantics>

</math><img src="./1003afab6a1efe9282d439ecd8bd941fd1797f95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.071ex; height:2.843ex;" alt="{\displaystyle \qquad \qquad (4.8)}" loading="lazy">

</div>

where

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {u} _{j}=\mathbf {L} (\mathbf {P} _{p,q}(2^{-\kappa _{j}}\mathbf {M} _{n}c_{j}h_{n}))^{2^{\kappa _{j}}}\mathbf {r} }</annotation>

</semantics>

</math><img src="./16218f30a65f01a92ff594c83ec074823e00abba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.157ex; height:3.676ex;" alt="{\displaystyle \mathbf {u} _{j}=\mathbf {L} (\mathbf {P} _{p,q}(2^{-\kappa _{j}}\mathbf {M} _{n}c_{j}h_{n}))^{2^{\kappa _{j}}}\mathbf {r} }" loading="lazy">

</div>

and

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {k} _{j}=\mathbf {f} \left(t_{n}+c_{j}h_{n},\mathbf {y} _{n}+\mathbf {u} _{j}+c_{j}h_{n}\mathbf {k} _{j-1}\right)-\mathbf {f} \left(t_{n},\mathbf {y} _{n}\right)-\mathbf {f} _{\mathbf {x} }\left(t_{n},\mathbf {y} _{n}\right)\mathbf {u} _{j}\ -\mathbf {f} _{t}\left(t_{n},\mathbf {y} _{n}\right)c_{j}h_{n},}</annotation>

</semantics>

</math><img src="./0fac0eced5021898f58e39d62340042a78c2baeb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:84.563ex; height:3.009ex;" alt="{\displaystyle \mathbf {k} _{j}=\mathbf {f} \left(t_{n}+c_{j}h_{n},\mathbf {y} _{n}+\mathbf {u} _{j}+c_{j}h_{n}\mathbf {k} _{j-1}\right)-\mathbf {f} \left(t_{n},\mathbf {y} _{n}\right)-\mathbf {f} _{\mathbf {x} }\left(t_{n},\mathbf {y} _{n}\right)\mathbf {u} _{j}\ -\mathbf {f} _{t}\left(t_{n},\mathbf {y} _{n}\right)c_{j}h_{n},}" loading="lazy">

</div>

with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {k} _{1}\equiv \mathbf {0} ,c=\left[{\begin{array}{cccc}0&{\frac {1}{2}}&{\frac {1}{2}}&1\end{array}}\right],}">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

</mfrac>

</mrow>

</mtd>

<mtd>

<mfrac>

</mfrac>

</mrow>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {k} _{1}\equiv \mathbf {0} ,c=\left[{\begin{array}{cccc}0&{\frac {1}{2}}&{\frac {1}{2}}&1\end{array}}\right],}</annotation>

</semantics>

</math><img src="./5c7569810b131c3d8e9ad24b6201f9454d90fbbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.968ex; margin-bottom: -0.203ex; width:28.373ex; height:3.509ex;" alt="{\displaystyle \mathbf {k} _{1}\equiv \mathbf {0} ,c=\left[{\begin{array}{cccc}0&{\frac {1}{2}}&{\frac {1}{2}}&1\end{array}}\right],}" loading="lazy"> and p + q > 3. For large systems of ODEs, the vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{j}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {u} _{j}}</annotation>

</semantics>

</math><img src="./0b494bd3864ed177c065a80b37f5ba19c2251fbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.395ex; height:2.343ex;" alt="{\displaystyle \mathbf {u} _{j}}" loading="lazy"> in the above scheme is replaced by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{j}=\mathbf {L\mathbf {k} } _{m_{j},k_{j}}^{p,q}(c_{j}h_{n},\mathbf {M} _{n},\mathbf {r} )}">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {u} _{j}=\mathbf {L\mathbf {k} } _{m_{j},k_{j}}^{p,q}(c_{j}h_{n},\mathbf {M} _{n},\mathbf {r} )}</annotation>

</semantics>

</math><img src="./eff1940d020487b28658c1322fdb909984a90936.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:26.139ex; height:3.843ex;" alt="{\displaystyle \mathbf {u} _{j}=\mathbf {L\mathbf {k} } _{m_{j},k_{j}}^{p,q}(c_{j}h_{n},\mathbf {M} _{n},\mathbf {r} )}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{j}>4.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m_{j}>4.}</annotation>

</semantics>

</math><img src="./7ffd195513039d9a4367fbeb883808d145b94c2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.858ex; height:2.843ex;" alt="{\displaystyle m_{j}>4.}" loading="lazy">

<div class="mw-heading mw-heading4"><h4 id="Locally_linearized_Runge–Kutta_scheme_of_Dormand_and_Prince">Locally linearized Runge–Kutta scheme of Dormand and Prince</h4></div>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{n+1}=\mathbf {y} _{n}+\mathbf {u} _{s}+h_{n}\sum _{j=1}^{s}b_{j}\mathbf {k} _{j}\qquad {\text{ and }}\qquad {\widehat {\mathbf {y} }}_{n+1}=\mathbf {y} _{n}+\mathbf {u} _{s}+h_{n}\sum _{j=1}^{s}{\widehat {b}}_{j}\mathbf {k} _{j},\quad }</annotation>

</semantics>

</math><img src="./dcb6da074ffe6590a172c4cc98dd0dffa711a092.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:76.417ex; height:7.176ex;" alt="{\displaystyle \mathbf {y} _{n+1}=\mathbf {y} _{n}+\mathbf {u} _{s}+h_{n}\sum _{j=1}^{s}b_{j}\mathbf {k} _{j}\qquad {\text{ and }}\qquad {\widehat {\mathbf {y} }}_{n+1}=\mathbf {y} _{n}+\mathbf {u} _{s}+h_{n}\sum _{j=1}^{s}{\widehat {b}}_{j}\mathbf {k} _{j},\quad }" loading="lazy">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \qquad \qquad (4.9)}</annotation>

</semantics>

</math><img src="./61cae1b45bf04e964778f4d5eca6cbafa0848067.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.071ex; height:2.843ex;" alt="{\displaystyle \qquad \qquad (4.9)}" loading="lazy">

</div>

where s = 7 is the number of stages,

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {k} _{j}=\mathbf {f(} t_{n}+c_{j}h_{n},\mathbf {y} _{n}+\mathbf {u} _{j}+h_{n}\sum _{i=1}^{s-1}a_{j,i}\mathbf {k} _{i})-\mathbf {f} \left(t_{n},\mathbf {y} _{n}\right)-\mathbf {f} _{\mathbf {x} }\left(t_{n},\mathbf {y} _{n}\right)\mathbf {u} _{j}\ -\mathbf {f} _{t}\left(t_{n},\mathbf {y} _{n}\right)c_{j}h_{n},}</annotation>

</semantics>

</math><img src="./81f021ebc979f5f9f3ade971d38f78df3be96b41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:87.477ex; height:7.343ex;" alt="{\displaystyle \mathbf {k} _{j}=\mathbf {f(} t_{n}+c_{j}h_{n},\mathbf {y} _{n}+\mathbf {u} _{j}+h_{n}\sum _{i=1}^{s-1}a_{j,i}\mathbf {k} _{i})-\mathbf {f} \left(t_{n},\mathbf {y} _{n}\right)-\mathbf {f} _{\mathbf {x} }\left(t_{n},\mathbf {y} _{n}\right)\mathbf {u} _{j}\ -\mathbf {f} _{t}\left(t_{n},\mathbf {y} _{n}\right)c_{j}h_{n},}" loading="lazy">

</div>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {k} _{1}\equiv \mathbf {0} }</annotation>

</semantics>

</math><img src="./46c2451cd7bbc623ad899557eefe7c42b10c4ccb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.9ex; height:2.509ex;" alt="{\displaystyle \mathbf {k} _{1}\equiv \mathbf {0} }" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{j,i},b_{j},{\widehat {b}}_{j}\quad and\quad c_{j}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{j,i},b_{j},{\widehat {b}}_{j}\quad and\quad c_{j}}</annotation>

</semantics>

</math><img src="./bdf505b7b14bf7ec83ebbf3cff735938efbcd915.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.614ex; height:3.509ex;" alt="{\displaystyle a_{j,i},b_{j},{\widehat {b}}_{j}\quad and\quad c_{j}}" loading="lazy"> are the <a href="Dormand%E2%80%93Prince_method" title="Dormand–Prince method">Runge–Kutta coefficients of Dormand and Prince</a> and p + q > 4. The vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{j}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {u} _{j}}</annotation>

</semantics>

<div class="mw-heading mw-heading4"><h4 id="Stability_and_dynamics">Stability and dynamics</h4></div>

By construction, the LL and HOLL discretizations inherit the stability and dynamics of the linear ODEs, but it is not the case of the LL schemes in general. With <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\leq q\leq p+2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\leq q\leq p+2}</annotation>

</semantics>

</math><img src="./fdc7e589b60294374f26e615dbcba5d137684a65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:13.698ex; height:2.509ex;" alt="{\displaystyle p\leq q\leq p+2}" loading="lazy">, the LL schemes (4.6)-(4.9) are <a href="Stiff_equation" title="Stiff equation">A-stable</a>.<a href="#cite_note-:3-4">[4]</a> With q = p + 1 or q = p + 2, the LL schemes (4.6)–(4.9) are also <a href="L-stability" title="L-stability">L-stable</a>.<a href="#cite_note-:3-4">[4]</a> For linear ODEs, the LL schemes (4.6)-(4.9) converge with order p + q.<a href="#cite_note-:3-4">[4]</a><a href="#cite_note-:25-9">[9]</a> In addition, with p = q = 6 and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m_{n}}</annotation>

</semantics>

</math><img src="./ae8181c5cf15da902bbaa5c2291aeb8167fe78ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.259ex; height:2.009ex;" alt="{\displaystyle m_{n}}" loading="lazy"> = d, all the above described LL schemes yield to the ″exact computation″ (up to the precision of the <a href="Floating-point_arithmetic" title="Floating-point arithmetic">floating-point arithmetic</a>) of linear ODEs on the current personal computers.<a href="#cite_note-:3-4">[4]</a><a href="#cite_note-:25-9">[9]</a> This includes <a href="Stiff_equation" title="Stiff equation">stiff</a> and highly oscillatory linear equations. Moreover, the LL schemes (4.6)-(4.9) are regular for linear ODEs and inherit the <a href="Symplectic_geometry" title="Symplectic geometry">symplectic structure</a> of <a href="Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian</a> <a href="Harmonic_oscillator" title="Harmonic oscillator">harmonic oscillators</a>.<a href="#cite_note-:2-5">[5]</a><a href="#cite_note-:9-13">[13]</a> These LL schemes are also linearization preserving, and display a better reproduction of the <a href="Stable_manifold" title="Stable manifold">stable and unstable manifolds</a> around <a href="Hyperbolic_equilibrium_point" title="Hyperbolic equilibrium point">hyperbolic equilibrium points</a> and <a href="Limit_cycle" title="Limit cycle">periodic orbits</a> that <a href="Numerical_methods_for_ordinary_differential_equations" title="Numerical methods for ordinary differential equations">other numerical schemes</a> with the same stepsize.<a href="#cite_note-:2-5">[5]</a><a href="#cite_note-:9-13">[13]</a> For instance, Figure 1 shows the <a href="Phase_portrait" title="Phase portrait">phase portrait</a> of the ODEs

<mtr>

<mtd>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\frac {dx_{1}}{dt}}=-2x_{1}+x_{2}+1-\mu f(x_{1},\lambda )\qquad \qquad (4.10)\\[6pt]&{\frac {dx_{2}}{dt}}=x_{1}-2x_{2}+1-\mu f(x_{2},\lambda )\qquad \qquad \quad (4.11)\end{aligned}}}</annotation>

</semantics>

</math><img src="./855ec9f888ae9f2a03221800414cbf76ef9edd46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:50.719ex; height:12.343ex;" alt="{\displaystyle {\begin{aligned}&{\frac {dx_{1}}{dt}}=-2x_{1}+x_{2}+1-\mu f(x_{1},\lambda )\qquad \qquad (4.10)\\[6pt]&{\frac {dx_{2}}{dt}}=x_{1}-2x_{2}+1-\mu f(x_{2},\lambda )\qquad \qquad \quad (4.11)\end{aligned}}}" loading="lazy"></dd></dl>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(u,\lambda )=u(1+u+\lambda u^{2})^{-1}}</annotation>

</semantics>

</math><img src="./2969d1be908ee3eb8d49c34b983b568ca715ae41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.289ex; height:3.176ex;" alt="{\displaystyle f(u,\lambda )=u(1+u+\lambda u^{2})^{-1}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =15}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu =15}</annotation>

</semantics>

</math><img src="./adff41daef86b8319734c944db5ba937c2aa56ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.825ex; height:2.676ex;" alt="{\displaystyle \mu =15}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =57}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda =57}</annotation>

</semantics>

</math><img src="./f8fee9544eaf9cddd5f2b374e990b5be64711851.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.779ex; height:2.176ex;" alt="{\displaystyle \lambda =57}" loading="lazy">, and its approximation by various schemes. This system has two <a href="Equilibrium_point" class="mw-redirect" title="Equilibrium point">stable stationary points</a> and one <a href="Equilibrium_point" class="mw-redirect" title="Equilibrium point">unstable stationary point</a> in the region <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq x_{1},x_{2}\leq 1}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\leq x_{1},x_{2}\leq 1}</annotation>

</semantics>

</math><img src="./58f5ba66529d7429c1dbd5dbd7892f70c566b94a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.324ex; height:2.509ex;" alt="{\displaystyle 0\leq x_{1},x_{2}\leq 1}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="LL_methods_for_DDEs">LL methods for DDEs</h2></div>

Consider the d-dimensional <a href="Delay_differential_equation" title="Delay differential equation">Delay Differential Equation</a> (DDE)

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {d\mathbf {x} (t)}{dt}}=\mathbf {f} (t,\mathbf {x} (t),\mathbf {x} _{t}(-\tau _{1}),\ldots ,\mathbf {x} _{t}(-\tau _{m})),\qquad t\in [t_{0},T],\qquad \qquad (5.1)}</annotation>

</semantics>

</math><img src="./682b3725e57d7e2ee5324a0a139b7e2d1063767b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:71.011ex; height:5.843ex;" alt="{\displaystyle {\frac {d\mathbf {x} (t)}{dt}}=\mathbf {f} (t,\mathbf {x} (t),\mathbf {x} _{t}(-\tau _{1}),\ldots ,\mathbf {x} _{t}(-\tau _{m})),\qquad t\in [t_{0},T],\qquad \qquad (5.1)}" loading="lazy">

</div>

with m constant delays <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{i}>0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau _{i}>0}</annotation>

</semantics>

</math><img src="./9f9d0c421128d29a40cc9a20d61bed4f16a42889.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.077ex; height:2.509ex;" alt="{\displaystyle \tau _{i}>0}" loading="lazy"> and initial condition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} _{t_{0}}(s)=\mathbf {\varphi } (s)}">

<msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{t_{0}}(s)=\mathbf {\varphi } (s)}</annotation>

</semantics>

</math><img src="./f80ff4d0afb417a9eb96ae047c4b4fb231f66a75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.487ex; height:3.009ex;" alt="{\displaystyle \mathbf {x} _{t_{0}}(s)=\mathbf {\varphi } (s)}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in [-\tau ,0],}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s\in [-\tau ,0],}</annotation>

</semantics>

</math><img src="./9e9d0232ea2938ec04dafc8665c88403f10670c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.078ex; height:2.843ex;" alt="{\displaystyle s\in [-\tau ,0],}" loading="lazy"> where f is a differentiable function, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} _{t}:[-\tau ,0]\longrightarrow \mathbb {R} ^{d}}">

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{t}:[-\tau ,0]\longrightarrow \mathbb {R} ^{d}}</annotation>

</semantics>

</math><img src="./026aa256631bc0e92437d8cea3a76cd0c6c2aded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.54ex; height:3.176ex;" alt="{\displaystyle \mathbf {x} _{t}:[-\tau ,0]\longrightarrow \mathbb {R} ^{d}}" loading="lazy"> is the segment function defined as

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{t}(s):=\mathbf {x} (t+s),{\text{ }}s\in [-\tau ,0],}</annotation>

</semantics>

</math><img src="./857fd392ec867a13d23dcb2ea2d705d132a706d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.565ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} _{t}(s):=\mathbf {x} (t+s),{\text{ }}s\in [-\tau ,0],}" loading="lazy">

</div>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t\in [t_{0},T],\mathbf {\varphi } :[-\tau ,0]\longrightarrow \mathbb {R} ^{d}}</annotation>

</semantics>

</math><img src="./332fbe38ea2b9aa1ca97db545170bba5e188883f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.395ex; height:3.176ex;" alt="{\displaystyle t\in [t_{0},T],\mathbf {\varphi } :[-\tau ,0]\longrightarrow \mathbb {R} ^{d}}" loading="lazy"> is a given function, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau =\max \left\{\tau _{1},\ldots ,\tau _{m}\right\}.}">

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau =\max \left\{\tau _{1},\ldots ,\tau _{m}\right\}.}</annotation>

</semantics>

</math><img src="./ba45f98733e0f769675b83cfe7d37976936a6d92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.312ex; height:2.843ex;" alt="{\displaystyle \tau =\max \left\{\tau _{1},\ldots ,\tau _{m}\right\}.}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Local_linear_discretization_2">Local linear discretization</h3></div>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (t)_{h}}</annotation>

</semantics>

</math><img src="./34cd9464ea6b4b6e27c0e854836a2840b3b24775.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.828ex; height:2.843ex;" alt="{\displaystyle (t)_{h}}" loading="lazy"> , the Local Linear discretization of the DDE (5.1) at each point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{n+1}\in (t)_{h}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{n+1}\in (t)_{h}}</annotation>

</semantics>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} _{n+1}=\mathbf {z} _{n}+\Phi (t_{n},\mathbf {z} _{n},h_{n};{\widetilde {\mathbf {z} }}_{t_{n}}^{1},\ldots ,{\widetilde {\mathbf {z} }}_{t_{n}}^{m}),\qquad \qquad (5.2)}</annotation>

</semantics>

</math><img src="./ddcff5429ba492b72bf323c6a4410b194951d31a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:52.705ex; height:3.343ex;" alt="{\displaystyle \mathbf {z} _{n+1}=\mathbf {z} _{n}+\Phi (t_{n},\mathbf {z} _{n},h_{n};{\widetilde {\mathbf {z} }}_{t_{n}}^{1},\ldots ,{\widetilde {\mathbf {z} }}_{t_{n}}^{m}),\qquad \qquad (5.2)}" loading="lazy">

</div>

where

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</munderover>

<msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<mrow>

<mrow>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</mrow>

</msubsup>

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</munderover>

</mrow>

</mrow>

</munderover>

<msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi (t_{n},\mathbf {z} _{n},h_{n};{\widetilde {\mathbf {z} }}_{t_{n}}^{1},\ldots ,{\widetilde {\mathbf {z} }}_{t_{n}}^{m})=\int \limits _{0}^{h_{n}}e^{\mathbf {A} _{n}(h_{n}-u)}\left[\sum \limits _{i=1}^{m}\mathbf {B} _{n}^{i}({\widetilde {\mathbf {z} }}_{t_{n}}^{i}(u-\tau _{i})-{\widetilde {\mathbf {z} }}_{t_{n}}^{i}(-\tau _{i}))+\mathbf {d} _{n}\right]\,du+\int \limits _{0}^{h_{n}}\int \limits _{0}^{u}e^{\mathbf {A} _{n}(h_{n}-u)}\mathbf {c} _{n}\,dr\,du}</annotation>

</semantics>

</math><img src="./d522057eb0209091646bcc9221b24d9e13d0e371.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:108.205ex; height:9.343ex;" alt="{\displaystyle \Phi (t_{n},\mathbf {z} _{n},h_{n};{\widetilde {\mathbf {z} }}_{t_{n}}^{1},\ldots ,{\widetilde {\mathbf {z} }}_{t_{n}}^{m})=\int \limits _{0}^{h_{n}}e^{\mathbf {A} _{n}(h_{n}-u)}\left[\sum \limits _{i=1}^{m}\mathbf {B} _{n}^{i}({\widetilde {\mathbf {z} }}_{t_{n}}^{i}(u-\tau _{i})-{\widetilde {\mathbf {z} }}_{t_{n}}^{i}(-\tau _{i}))+\mathbf {d} _{n}\right]\,du+\int \limits _{0}^{h_{n}}\int \limits _{0}^{u}e^{\mathbf {A} _{n}(h_{n}-u)}\mathbf {c} _{n}\,dr\,du}" loading="lazy">

</div>

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widetilde {\mathbf {z} }}_{t_{n}}^{i}:\left[-\tau _{i},0\right]\longrightarrow \mathbb {R} ^{d}}</annotation>

</semantics>

</math><img src="./d3f4bc0cc5b10a10d0844d2265aababd5028d6e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.896ex; height:3.509ex;" alt="{\displaystyle {\widetilde {\mathbf {z} }}_{t_{n}}^{i}:\left[-\tau _{i},0\right]\longrightarrow \mathbb {R} ^{d}}" loading="lazy"> is the segment function defined as

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msup>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widetilde {\mathbf {z} }}_{t_{n}}^{i}(s):={\widetilde {\mathbf {z} }}^{i}(t_{n}+s),{\text{ }}s\in [-\tau _{i},0],}</annotation>

</semantics>

</math><img src="./d4c1d308208faf6bc44261b2ce1b128a9f394195.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:32.716ex; height:3.509ex;" alt="{\displaystyle {\widetilde {\mathbf {z} }}_{t_{n}}^{i}(s):={\widetilde {\mathbf {z} }}^{i}(t_{n}+s),{\text{ }}s\in [-\tau _{i},0],}" loading="lazy">

</div>

and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {\mathbf {z} }}^{i}:\left[t_{n}-\tau _{i},t_{n}\right]\longrightarrow \mathbb {R} ^{d}}">

<msup>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</msup>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widetilde {\mathbf {z} }}^{i}:\left[t_{n}-\tau _{i},t_{n}\right]\longrightarrow \mathbb {R} ^{d}}</annotation>

</semantics>

</math><img src="./922ac10d7369b3ff0ffe9487ebb8a509a1305574.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.891ex; height:3.176ex;" alt="{\displaystyle {\widetilde {\mathbf {z} }}^{i}:\left[t_{n}-\tau _{i},t_{n}\right]\longrightarrow \mathbb {R} ^{d}}" loading="lazy"> is a suitable approximation to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} (t)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} (t)}</annotation>

</semantics>

</math><img src="./385450fed781e07652f160faa271a7cc17fdcbc0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.06ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} (t)}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in \lbrack t_{n}-\tau _{i},t_{n}]}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t\in \lbrack t_{n}-\tau _{i},t_{n}]}</annotation>

</semantics>

</math><img src="./1f287009ce5d61c9d7ed2a843c93dfdef6226a50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.78ex; height:2.843ex;" alt="{\displaystyle t\in \lbrack t_{n}-\tau _{i},t_{n}]}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {\mathbf {z} }}^{i}(t_{n})=\mathbf {z} _{n}.}">

<msup>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widetilde {\mathbf {z} }}^{i}(t_{n})=\mathbf {z} _{n}.}</annotation>

</semantics>

</math><img src="./fe4b4fc9d49682507c04341cb3d19057d59ba06b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.007ex; height:3.176ex;" alt="{\displaystyle {\widetilde {\mathbf {z} }}^{i}(t_{n})=\mathbf {z} _{n}.}" loading="lazy"> Here,<div style="text-align: center;">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} _{n}=\mathbf {f} _{x}(t_{n},\mathbf {z} _{n},{\widetilde {\mathbf {z} }}_{t_{n}}^{1}(-\tau _{1}),\ldots ,{\widetilde {\mathbf {z} }}_{t_{n}}^{m}(-\tau _{d})),{\text{ }}\mathbf {B} _{n}^{i}=\mathbf {f} _{x_{t}(-\tau _{i})}(t_{n},\mathbf {z} _{n},{\widetilde {\mathbf {z} }}_{t_{n}}^{1}(-\tau _{1}),\ldots ,{\widetilde {\mathbf {z} }}_{t_{n}}^{m}(-\tau _{d}))}</annotation>

</semantics>

</math><img src="./c1267fedbd8c87f1f520740cf2fc3ab5ffeb1a26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:84.476ex; height:3.509ex;" alt="{\displaystyle \mathbf {A} _{n}=\mathbf {f} _{x}(t_{n},\mathbf {z} _{n},{\widetilde {\mathbf {z} }}_{t_{n}}^{1}(-\tau _{1}),\ldots ,{\widetilde {\mathbf {z} }}_{t_{n}}^{m}(-\tau _{d})),{\text{ }}\mathbf {B} _{n}^{i}=\mathbf {f} _{x_{t}(-\tau _{i})}(t_{n},\mathbf {z} _{n},{\widetilde {\mathbf {z} }}_{t_{n}}^{1}(-\tau _{1}),\ldots ,{\widetilde {\mathbf {z} }}_{t_{n}}^{m}(-\tau _{d}))}" loading="lazy">

</div>

are constant matrices and

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {c} _{n}=\mathbf {f} _{t}(t_{n},\mathbf {z} _{n},{\widetilde {\mathbf {z} }}_{t_{n}}^{1}(-\tau _{1}),\ldots ,{\widetilde {\mathbf {z} }}_{t_{n}}^{m}(-\tau _{d})){\text{ and }}\mathbf {d} _{n}=\mathbf {f(} t_{n},\mathbf {z} _{n},{\widetilde {\mathbf {z} }}_{t_{n}}^{1}(-\tau _{1}),\ldots ,{\widetilde {\mathbf {z} }}_{t_{n}}^{m}(-\tau _{d}))}</annotation>

</semantics>

</math><img src="./0e5d089f728f55946c6dd8cf13d277af20dfea21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:80.828ex; height:3.343ex;" alt="{\displaystyle \mathbf {c} _{n}=\mathbf {f} _{t}(t_{n},\mathbf {z} _{n},{\widetilde {\mathbf {z} }}_{t_{n}}^{1}(-\tau _{1}),\ldots ,{\widetilde {\mathbf {z} }}_{t_{n}}^{m}(-\tau _{d})){\text{ and }}\mathbf {d} _{n}=\mathbf {f(} t_{n},\mathbf {z} _{n},{\widetilde {\mathbf {z} }}_{t_{n}}^{1}(-\tau _{1}),\ldots ,{\widetilde {\mathbf {z} }}_{t_{n}}^{m}(-\tau _{d}))}" loading="lazy">

</div>

are constant vectors. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} _{t},\mathbf {f} _{x}\quad and\quad \mathbf {f} _{x_{t}(-\tau _{i})}}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {f} _{t},\mathbf {f} _{x}\quad and\quad \mathbf {f} _{x_{t}(-\tau _{i})}}</annotation>

</semantics>

</math><img src="./b0fd7595f389635c01934d6984ca9e85cfed92c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.687ex; height:3.009ex;" alt="{\displaystyle \mathbf {f} _{t},\mathbf {f} _{x}\quad and\quad \mathbf {f} _{x_{t}(-\tau _{i})}}" loading="lazy"> denote, respectively, the partial derivatives of f with respect to the variables t and x, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} _{t}(-\tau _{i})}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{t}(-\tau _{i})}</annotation>

</semantics>

</math><img src="./c47846ed104e2cbe210c6bb8c78b67d71b839561.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.67ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} _{t}(-\tau _{i})}" loading="lazy">. The Local Linear discretization (5.2) converges to the solution of (5.1) with order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\min\{2,r\},}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha =\min\{2,r\},}</annotation>

</semantics>

</math><img src="./6fbb5633ea5155b6ae7c31f9a348761bd1ebc5fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.678ex; height:2.843ex;" alt="{\displaystyle \alpha =\min\{2,r\},}" loading="lazy"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {\mathbf {z} }}_{t_{n}}^{i}}">

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widetilde {\mathbf {z} }}_{t_{n}}^{i}}</annotation>

</semantics>

</math><img src="./8f938c4f67e8dc674c56ecfecd642a238c6cb09a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.979ex; height:3.343ex;" alt="{\displaystyle {\widetilde {\mathbf {z} }}_{t_{n}}^{i}}" loading="lazy"> approximates <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {z} _{t_{n}}^{i}}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} _{t_{n}}^{i}}</annotation>

</semantics>

</math><img src="./ff8a7a8aca1698ca90d10d0497fc3f1d30d7cbfd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.979ex; height:3.343ex;" alt="{\displaystyle \mathbf {z} _{t_{n}}^{i}}" loading="lazy"> with order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\quad (i.e.,\left\vert \mathbf {z} _{t_{n}}^{i}\mathbf {(} u-\tau _{i}\mathbf {)} -{\widetilde {\mathbf {z} }}_{t_{n}}^{i}\mathbf {(} u-\tau _{i}\mathbf {)} \right\vert \propto h_{n}^{r}}">

<mrow>

<mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r\quad (i.e.,\left\vert \mathbf {z} _{t_{n}}^{i}\mathbf {(} u-\tau _{i}\mathbf {)} -{\widetilde {\mathbf {z} }}_{t_{n}}^{i}\mathbf {(} u-\tau _{i}\mathbf {)} \right\vert \propto h_{n}^{r}}</annotation>

</semantics>

</math><img src="./759e22f17bd252643f82f89bb9834dd9174a5f44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:41.141ex; height:3.509ex;" alt="{\displaystyle r\quad (i.e.,\left\vert \mathbf {z} _{t_{n}}^{i}\mathbf {(} u-\tau _{i}\mathbf {)} -{\widetilde {\mathbf {z} }}_{t_{n}}^{i}\mathbf {(} u-\tau _{i}\mathbf {)} \right\vert \propto h_{n}^{r}}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in \lbrack 0,h_{n}])}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u\in \lbrack 0,h_{n}])}</annotation>

</semantics>

</math><img src="./956d5d79b8e61251940a26652b1ecead7daa8b6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.123ex; height:2.843ex;" alt="{\displaystyle u\in \lbrack 0,h_{n}])}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Local_linearization_schemes_2">Local linearization schemes</h3></div>

Depending on the approximations <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {\mathbf {z} }}_{t_{n}}^{i}}">

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widetilde {\mathbf {z} }}_{t_{n}}^{i}}</annotation>

</semantics>

</math><img src="./8f938c4f67e8dc674c56ecfecd642a238c6cb09a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.979ex; height:3.343ex;" alt="{\displaystyle {\widetilde {\mathbf {z} }}_{t_{n}}^{i}}" loading="lazy"> and on the algorithm to compute <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\phi } }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\phi } }</annotation>

</semantics>

</math><img src="./03b16c8b97a15009afffb576b1424e184f48f6b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \mathbf {\phi } }" loading="lazy"> different Local Linearizations schemes can be defined. Every numerical implementation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} _{n}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{n}}</annotation>

</semantics>

</math><img src="./05275acac23ef658db9a5d0b777828444fa2096c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.629ex; height:2.176ex;" alt="{\displaystyle \mathbf {y} _{n}}" loading="lazy"> of a Local Linear discretization <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {z} _{n}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} _{n}}</annotation>

</semantics>

<div class="mw-heading mw-heading4"><h4 id="Order-2_polynomial_LL_schemes">Order-2 polynomial LL schemes</h4></div>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

</semantics>

</math><img src="./190b3e73891ef32cbdcf5adc1c94f3cda8b86dbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.447ex; height:3.676ex;" alt="{\displaystyle \mathbf {y} _{n+1}=\mathbf {y} _{n}+\mathbf {L} (\mathbf {P} _{p,q}(2^{-k_{n}}\mathbf {M} _{n}h_{n}))^{2^{k_{n}}}\mathbf {r,} \quad }" loading="lazy"><a href="#cite_note-:13-11">[11]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad (5.3)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \qquad (5.3)}</annotation>

</semantics>

</math><img src="./4610e90de804280c0ca7dba13be1dc3abd57af4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.426ex; height:2.843ex;" alt="{\displaystyle \qquad (5.3)}" loading="lazy">

</div>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {M} _{n},\mathbf {L} }</annotation>

</semantics>

</math><img src="./05286927c0e23c53706cb8014b6b7e7eead127d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.398ex; height:2.509ex;" alt="{\displaystyle \mathbf {M} _{n},\mathbf {L} }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation>

</semantics>

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {M} _{n}={\begin{bmatrix}\mathbf {A} _{n}&\mathbf {c} _{n}+\sum \limits _{i=1}^{m}\mathbf {B} _{n}^{i}\mathbf {\alpha } _{n}^{i}&\mathbf {d} _{n}\\0&0&1\\0&0&0\end{bmatrix}}\in \mathbb {R} ^{(d+2)\times (d+2)},}</annotation>

</semantics>

</math><img src="./ae5bbe4975b5e5690241b3fa9b3d1b54fcaa6924.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:50.577ex; height:12.509ex;" alt="{\displaystyle \mathbf {M} _{n}={\begin{bmatrix}\mathbf {A} _{n}&\mathbf {c} _{n}+\sum \limits _{i=1}^{m}\mathbf {B} _{n}^{i}\mathbf {\alpha } _{n}^{i}&\mathbf {d} _{n}\\0&0&1\\0&0&0\end{bmatrix}}\in \mathbb {R} ^{(d+2)\times (d+2)},}" loading="lazy">

</div>

</mrow>

<mrow>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {L} =\left[{\begin{array}{ll}\mathbf {I} &\mathbf {0} _{d\times 2}\end{array}}\right]}</annotation>

</semantics>

<msup>

</mrow>

</mrow>

</msup>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} ^{\intercal }=\left[{\begin{array}{ll}\mathbf {0} _{1\times (d+1)}&1\end{array}}\right],h_{n}\leq \tau }</annotation>

</semantics>

</math><img src="./05fdb2050667c7b12025db96c59b015211204a57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.711ex; height:3.176ex;" alt="{\displaystyle \mathbf {r} ^{\intercal }=\left[{\begin{array}{ll}\mathbf {0} _{1\times (d+1)}&1\end{array}}\right],h_{n}\leq \tau }" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p+q>1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p+q>1}</annotation>

</semantics>

</math><img src="./5340225ba272f9ca32d1fc02c246d11737fed65c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.43ex; height:2.509ex;" alt="{\displaystyle p+q>1}" loading="lazy">. Here, the matrices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} _{n}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} _{n}}</annotation>

</semantics>

</math><img src="./16a6914d0625628c2c524d7c0bd484a7eeaa601a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.238ex; height:2.509ex;" alt="{\displaystyle \mathbf {A} _{n}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} _{n}^{i}}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {B} _{n}^{i}}</annotation>

</semantics>

</math><img src="./3017ac6e9a4d15e763d37a34d84f6a404a2b5a2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.12ex; height:3.176ex;" alt="{\displaystyle \mathbf {B} _{n}^{i}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {c} _{n}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {c} _{n}}</annotation>

</semantics>

</math><img src="./2df34d3c61b20a66dff4276f02e9c7a31652b632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.406ex; height:2.009ex;" alt="{\displaystyle \mathbf {c} _{n}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {d} _{n}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {d} _{n}}</annotation>

</semantics>

</math><img src="./14796912eb729a254cd14cc4647626097aae007f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.704ex; height:2.509ex;" alt="{\displaystyle \mathbf {d} _{n}}" loading="lazy"> are defined as in (5.2), but replacing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} }</annotation>

</semantics>

</math><img src="./82eca5d0928078d5a61b9e7e98cc73db31070909.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.188ex; height:1.676ex;" alt="{\displaystyle \mathbf {z} }" loading="lazy"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} }</annotation>

</semantics>

</math><img src="./bb25a040b592282dc2a254c3117e792c3c81161f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.411ex; height:2.009ex;" alt="{\displaystyle \mathbf {y} }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\alpha } _{n}^{i}=(\mathbf {y} (t_{n+1}-\tau _{i})-\mathbf {y} (t_{n}-\tau _{i}))/h_{n},}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\alpha } _{n}^{i}=(\mathbf {y} (t_{n+1}-\tau _{i})-\mathbf {y} (t_{n}-\tau _{i}))/h_{n},}</annotation>

</semantics>

</math><img src="./a6cbc7b6d6d9742aa8f615409dc35105e4198e8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.791ex; height:3.009ex;" alt="{\displaystyle \mathbf {\alpha } _{n}^{i}=(\mathbf {y} (t_{n+1}-\tau _{i})-\mathbf {y} (t_{n}-\tau _{i}))/h_{n},}" loading="lazy"> where

</mrow>

<mrow>

</mrow>

<msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} \left(t\right)=\mathbf {y} _{n_{t}}+\mathbf {L} (\mathbf {P} _{p,q}(2^{-k_{n}}\mathbf {M} _{n_{t}}(t-t_{n_{t}})))^{2^{k_{n}}}\mathbf {r} ,}</annotation>

</semantics>

</math><img src="./8b1cf4fe0b047bda0c58604a75247706f9c55f67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:44.675ex; height:3.843ex;" alt="{\displaystyle \mathbf {y} \left(t\right)=\mathbf {y} _{n_{t}}+\mathbf {L} (\mathbf {P} _{p,q}(2^{-k_{n}}\mathbf {M} _{n_{t}}(t-t_{n_{t}})))^{2^{k_{n}}}\mathbf {r} ,}" loading="lazy">

</div>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mrow>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n_{t}=\max\{n=0,1,2,...,:t_{n}\leq t{\text{ and }}t_{n}\in \left(t\right)_{h}\}}</annotation>

</semantics>

</math><img src="./1393cd8d04da6f35cc101582305866485ec08652.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:48.112ex; height:3.009ex;" alt="{\displaystyle n_{t}=\max\{n=0,1,2,...,:t_{n}\leq t{\text{ and }}t_{n}\in \left(t\right)_{h}\}}" loading="lazy">, is the Local Linear Approximation to the solution of (5.1) defined through the LL scheme (5.3) for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in \lbrack t_{0},t_{n}]}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t\in \lbrack t_{0},t_{n}]}</annotation>

</semantics>

</math><img src="./8ed17e1a454d79e9e85aa3fdb316727d17eb069b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.96ex; height:2.843ex;" alt="{\displaystyle t\in \lbrack t_{0},t_{n}]}" loading="lazy"> and by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} \left(t\right)=\mathbf {\varphi } \left(t\right)}">

</mrow>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} \left(t\right)=\mathbf {\varphi } \left(t\right)}</annotation>

</semantics>

</math><img src="./428e25fbaf900521af2eb6adb752e70b73fd18da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.102ex; height:2.843ex;" alt="{\displaystyle \mathbf {y} \left(t\right)=\mathbf {\varphi } \left(t\right)}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in \left[t_{0}-\tau ,t_{0}\right]}">

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t\in \left[t_{0}-\tau ,t_{0}\right]}</annotation>

</semantics>

</math><img src="./99fafe4165d02d310ce7c3f55c6fbfc1742d2d65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.838ex; height:2.843ex;" alt="{\displaystyle t\in \left[t_{0}-\tau ,t_{0}\right]}" loading="lazy">. For large systems of DDEs

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

</mrow>

<msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mrow>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{n+1}=\mathbf {y} _{n}+\mathbf {L\mathbf {k} } _{m_{n},k_{n}}^{p,q}(h_{n},\mathbf {M} _{n},\mathbf {r} )\quad and\quad \mathbf {y} \left(t\right)=\mathbf {y} _{n_{t}}+\mathbf {L\mathbf {k} } _{m_{n_{t}},k_{n_{t}}}^{p,q}(t-t_{n_{t}},\mathbf {M} _{n_{t}},\mathbf {r} ),}</annotation>

</semantics>

</math><img src="./94126e2f37056f2ba72b2caa673b3fafbba40bc2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:80.257ex; height:3.843ex;" alt="{\displaystyle \mathbf {y} _{n+1}=\mathbf {y} _{n}+\mathbf {L\mathbf {k} } _{m_{n},k_{n}}^{p,q}(h_{n},\mathbf {M} _{n},\mathbf {r} )\quad and\quad \mathbf {y} \left(t\right)=\mathbf {y} _{n_{t}}+\mathbf {L\mathbf {k} } _{m_{n_{t}},k_{n_{t}}}^{p,q}(t-t_{n_{t}},\mathbf {M} _{n_{t}},\mathbf {r} ),}" loading="lazy">

</div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p+q>1}</annotation>

</semantics>

</math><img src="./5340225ba272f9ca32d1fc02c246d11737fed65c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.43ex; height:2.509ex;" alt="{\displaystyle p+q>1}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{n}>2}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m_{n}>2}</annotation>

</semantics>

</math><img src="./94b0f083045b936b4a49670b6eece1084d68a244.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.52ex; height:2.509ex;" alt="{\displaystyle m_{n}>2}" loading="lazy">. Fig. 2 Illustrates the stability of the LL scheme (5.3) and of that of an explicit scheme of similar order in the integration of a stiff system of DDEs.

<div class="mw-heading mw-heading2"><h2 id="LL_methods_for_RDEs">LL methods for RDEs</h2></div>

Consider the d-dimensional Random Differential Equation (RDE)

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {d\mathbf {x} \left(t\right)}{dt}}=\mathbf {f} (\mathbf {x} (t),\mathbf {\xi } (t)),\quad t\in \left[t_{0},T\right],\qquad \qquad \qquad (6.1)}</annotation>

</semantics>

</math><img src="./06e7f4c5cbe530e829f8edf84e2e49d0829a2ef0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:54.265ex; height:5.843ex;" alt="{\displaystyle {\frac {d\mathbf {x} \left(t\right)}{dt}}=\mathbf {f} (\mathbf {x} (t),\mathbf {\xi } (t)),\quad t\in \left[t_{0},T\right],\qquad \qquad \qquad (6.1)}" loading="lazy">

</div>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} (t_{0})=\mathbf {x} _{0},}</annotation>

</semantics>

</math><img src="./d62793e41168b799ef6fade678d98ae11c112ec7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.325ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} (t_{0})=\mathbf {x} _{0},}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\xi } }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\xi } }</annotation>

</semantics>

</math><img src="./b2e2a51fb5522ad140f19673a76daef706190448.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.03ex; height:2.509ex;" alt="{\displaystyle \mathbf {\xi } }" loading="lazy"> is a k-dimensional <a href="Stochastic_process" title="Stochastic process">separable finite continuous stochastic process</a>, and f is a differentiable function. Suppose that a <a href="Realization_(probability)" title="Realization (probability)">realization</a> (path) of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\xi } }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\xi } }</annotation>

</semantics>

<div class="mw-heading mw-heading3"><h3 id="Local_Linear_discretization_3">Local Linear discretization</h3></div>

<msub>

<mrow>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(t\right)_{h}}</annotation>

</semantics>

</math><img src="./16c122098f93662ae323d6fe099721b11f92825b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.828ex; height:3.009ex;" alt="{\displaystyle \left(t\right)_{h}}" loading="lazy">, the Local Linear discretization of the RDE (6.1) at each point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{n+1}\in \left(t\right)_{h}}">

<msub>

</mrow>

</msub>

<msub>

<mrow>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{n+1}\in \left(t\right)_{h}}</annotation>

</semantics>

</math><img src="./59bd1e8eb52a954e56d94e1236bbcd920f78110d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.827ex; height:3.009ex;" alt="{\displaystyle t_{n+1}\in \left(t\right)_{h}}" loading="lazy"> is defined by the recursive expression <a href="#cite_note-:24-16">[16]</a>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} _{n+1}=\mathbf {z} _{n}+\mathbf {\phi } (t_{n},\mathbf {z} _{n};h_{n}),\qquad {\text{ with }}\qquad \mathbf {z} _{0}=\mathbf {x} _{0},}</annotation>

</semantics>

</math><img src="./b5b07593b82cdfda4d25ac671f0c10ee7c76ec2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.597ex; height:2.843ex;" alt="{\displaystyle \mathbf {z} _{n+1}=\mathbf {z} _{n}+\mathbf {\phi } (t_{n},\mathbf {z} _{n};h_{n}),\qquad {\text{ with }}\qquad \mathbf {z} _{0}=\mathbf {x} _{0},}" loading="lazy">

</div>

where

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</munderover>

<msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\phi } (t_{n},\mathbf {z} _{n};h_{n})=\int \limits _{0}^{h_{n}}e^{\mathbf {f} _{\mathbf {x} }(\mathbf {z} _{n},\mathbf {\xi } (t_{n}))(h_{n}-u)}(\mathbf {f(z} _{n},\mathbf {\xi } (t_{n}))+\mathbf {f} _{\mathbf {\xi } }(\mathbf {z} _{n},\mathbf {\xi } (t_{n}))({\widetilde {\mathbf {\xi } }}(t_{n}+u)-{\widetilde {\mathbf {\xi } }}(t_{n})))\,du}</annotation>

</semantics>

</math><img src="./3ef3cc9d64de2f89f3fc309973cb4fc8b2beeeaf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:84.046ex; height:9.343ex;" alt="{\displaystyle \mathbf {\phi } (t_{n},\mathbf {z} _{n};h_{n})=\int \limits _{0}^{h_{n}}e^{\mathbf {f} _{\mathbf {x} }(\mathbf {z} _{n},\mathbf {\xi } (t_{n}))(h_{n}-u)}(\mathbf {f(z} _{n},\mathbf {\xi } (t_{n}))+\mathbf {f} _{\mathbf {\xi } }(\mathbf {z} _{n},\mathbf {\xi } (t_{n}))({\widetilde {\mathbf {\xi } }}(t_{n}+u)-{\widetilde {\mathbf {\xi } }}(t_{n})))\,du}" loading="lazy">

</div>

and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {\mathbf {\xi } }}}">

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widetilde {\mathbf {\xi } }}}</annotation>

</semantics>

</math><img src="./77acc7c2b6a64b515827b1d31831fb3b4d15133f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.428ex; height:3.009ex;" alt="{\displaystyle {\widetilde {\mathbf {\xi } }}}" loading="lazy"> is an approximation to the process <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\xi } }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\xi } }</annotation>

</semantics>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t\in \left[t_{0},T\right].}</annotation>

</semantics>

</math><img src="./aedde214e225a352a24c1c922043d56a7d90588b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.572ex; height:2.843ex;" alt="{\displaystyle t\in \left[t_{0},T\right].}" loading="lazy"> Here, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} _{x}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {f} _{x}}</annotation>

</semantics>

</math><img src="./e831a0e34ab7d51df3c82602832ebe3444700ccb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.989ex; height:2.509ex;" alt="{\displaystyle \mathbf {f} _{x}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} _{\xi }}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {f} _{\xi }}</annotation>

</semantics>

</math><img src="./2c45ac3e742fef16bdf9c61f4cf88feae0293040.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.777ex; height:2.843ex;" alt="{\displaystyle \mathbf {f} _{\xi }}" loading="lazy"> denote the partial derivatives of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {f} }</annotation>

</semantics>

</math><img src="./dc6194e680a4e7c521f2178c50eea302843a852d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.053ex; height:2.176ex;" alt="{\displaystyle \mathbf {f} }" loading="lazy"> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation>

</semantics>

</math><img src="./32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \xi }</annotation>

</semantics>

</math><img src="./e0b461aaf61091abd5d2c808931c48b8ff9647db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.03ex; height:2.509ex;" alt="{\displaystyle \xi }" loading="lazy">, respectively.

<div class="mw-heading mw-heading3"><h3 id="Local_linearization_schemes_3">Local linearization schemes</h3></div>

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widetilde {\mathbf {\xi } }}}</annotation>

</semantics>

</math><img src="./77acc7c2b6a64b515827b1d31831fb3b4d15133f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.428ex; height:3.009ex;" alt="{\displaystyle {\widetilde {\mathbf {\xi } }}}" loading="lazy"> to the process <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\xi } }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\xi } }</annotation>

</semantics>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\phi } }</annotation>

</semantics>

</math><img src="./03b16c8b97a15009afffb576b1424e184f48f6b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \mathbf {\phi } }" loading="lazy">, different Local Linearizations schemes can be defined. Every numerical implementation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} _{n}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{n}}</annotation>

</semantics>

</math><img src="./05275acac23ef658db9a5d0b777828444fa2096c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.629ex; height:2.176ex;" alt="{\displaystyle \mathbf {y} _{n}}" loading="lazy"> of the local linear discretization <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {z} _{n}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} _{n}}</annotation>

</semantics>

<div class="mw-heading mw-heading4"><h4 id="LL_schemes">LL schemes</h4></div>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

</semantics>

</math><img src="./190b3e73891ef32cbdcf5adc1c94f3cda8b86dbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.447ex; height:3.676ex;" alt="{\displaystyle \mathbf {y} _{n+1}=\mathbf {y} _{n}+\mathbf {L} (\mathbf {P} _{p,q}(2^{-k_{n}}\mathbf {M} _{n}h_{n}))^{2^{k_{n}}}\mathbf {r,} \quad }" loading="lazy"> <a href="#cite_note-:24-16">[16]</a><a href="#cite_note-:10-17">[17]</a> </div>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {M} _{n},\quad \mathbf {L} \quad and\quad \mathbf {r} }</annotation>

</semantics>

</math><img src="./4c1fd09ea7d02b8cf7908751633c698164d4208c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.308ex; height:2.509ex;" alt="{\displaystyle \mathbf {M} _{n},\quad \mathbf {L} \quad and\quad \mathbf {r} }" loading="lazy"> are defined as

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {M} _{n}=\left[{\begin{array}{ccc}\mathbf {f} _{\mathbf {x} }\left(\mathbf {y} _{n},\mathbf {\xi } (t_{n})\right)&\mathbf {f} _{\mathbf {\xi } }(\mathbf {y} _{n},\mathbf {\xi } (t_{n})(\mathbf {\xi } (t_{n+1})-\mathbf {\xi } (t_{n}))/h_{n}&\mathbf {f} \left(\mathbf {y} _{n},\mathbf {\xi } (t_{n})\right)\\0&0&1\\0&0&0\end{array}}\right]}</annotation>

</semantics>

</math><img src="./de2e549c00febeadc80e1df68d44a6a66abd159a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:71.473ex; height:9.509ex;" alt="{\displaystyle \mathbf {M} _{n}=\left[{\begin{array}{ccc}\mathbf {f} _{\mathbf {x} }\left(\mathbf {y} _{n},\mathbf {\xi } (t_{n})\right)&\mathbf {f} _{\mathbf {\xi } }(\mathbf {y} _{n},\mathbf {\xi } (t_{n})(\mathbf {\xi } (t_{n+1})-\mathbf {\xi } (t_{n}))/h_{n}&\mathbf {f} \left(\mathbf {y} _{n},\mathbf {\xi } (t_{n})\right)\\0&0&1\\0&0&0\end{array}}\right]}" loading="lazy">

</mrow>

<mrow>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {L} =\left[{\begin{array}{ll}\mathbf {I} &\mathbf {0} _{d\times 2}\end{array}}\right]}</annotation>

</semantics>

</math><img src="./05d54f18477ecdb4b717abf25a75b96d17729bcd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.618ex; height:2.843ex;" alt="{\displaystyle \mathbf {L} =\left[{\begin{array}{ll}\mathbf {I} &\mathbf {0} _{d\times 2}\end{array}}\right]}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} ^{\intercal }=\left[{\begin{array}{ll}\mathbf {0} _{1\times (d+1)}&1\end{array}}\right]}">

<msup>

</mrow>

</mrow>

</msup>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} ^{\intercal }=\left[{\begin{array}{ll}\mathbf {0} _{1\times (d+1)}&1\end{array}}\right]}</annotation>

</semantics>

</math><img src="./f82230fb66e66705351377690e9f14c8d4a41224.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.432ex; height:3.176ex;" alt="{\displaystyle \mathbf {r} ^{\intercal }=\left[{\begin{array}{ll}\mathbf {0} _{1\times (d+1)}&1\end{array}}\right]}" loading="lazy">, and p+q>1. For large systems of RDEs,<a href="#cite_note-:10-17">[17]</a>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

</semantics>

</math><img src="./d9ad3b8df9f4a702b508729139a25818c2932b20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:61.878ex; height:3.509ex;" alt="{\displaystyle \mathbf {y} _{n+1}=\mathbf {y} _{n}+\mathbf {L\mathbf {k} } _{m_{n},k_{n}}^{p,q}(h_{n},\mathbf {M} _{n},\mathbf {r} ),\quad p+q>1\quad and\quad m_{n}>2.}" loading="lazy">

</div>

The convergence rate of both schemes is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle min\{2,2\gamma \}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle min\{2,2\gamma \}}</annotation>

</semantics>

</math><img src="./bb62d4259c5c9cfdc9e1b888c2aee135df9ac284.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.184ex; height:2.843ex;" alt="{\displaystyle min\{2,2\gamma \}}" loading="lazy">, where is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

</semantics>

</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> the exponent of the Holder condition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\xi } }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\xi } }</annotation>

</semantics>

Figure 3 presents the phase portrait of the RDE

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {dx_{1}}{dt}}=-x_{2}+\left(1-x_{1}^{2}-x_{2}^{2}\right)x_{1}\sin(w^{H}(t))^{2},\quad \qquad x_{1}(0)=0.8\qquad (6.2)}</annotation>

</semantics>

</math><img src="./06d6a9275659c3da93651ea621ca4d346d437ff2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:72.043ex; height:5.509ex;" alt="{\displaystyle {\frac {dx_{1}}{dt}}=-x_{2}+\left(1-x_{1}^{2}-x_{2}^{2}\right)x_{1}\sin(w^{H}(t))^{2},\quad \qquad x_{1}(0)=0.8\qquad (6.2)}" loading="lazy">

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {dx_{2}}{dt}}=x_{1}+(1-x_{1}^{2}-x_{2}^{2})x_{2}\sin(w^{H}(t))^{2},\qquad \qquad x_{2}(0)=0.1,\qquad (6.3)}</annotation>

</semantics>

</math><img src="./fed15f3afd9950f78676764274198ef8d7a70801.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:72.884ex; height:5.509ex;" alt="{\displaystyle {\frac {dx_{2}}{dt}}=x_{1}+(1-x_{1}^{2}-x_{2}^{2})x_{2}\sin(w^{H}(t))^{2},\qquad \qquad x_{2}(0)=0.1,\qquad (6.3)}" loading="lazy">

and its approximation by two numerical schemes, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w^{H}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w^{H}}</annotation>

</semantics>

</math><img src="./35da807bf9ff1af06231d51fc065762d5ffc7d2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.356ex; height:2.676ex;" alt="{\displaystyle w^{H}}" loading="lazy"> denotes a <a href="Fractional_Brownian_motion" title="Fractional Brownian motion">fractional Brownian process</a> with <a href="Hurst_exponent" title="Hurst exponent">Hurst exponent</a> H=0.45.

<div class="mw-heading mw-heading2"><h2 id="Strong_LL_methods_for_SDEs">Strong LL methods for SDEs</h2></div>

Consider the d-dimensional <a href="Stochastic_differential_equation" title="Stochastic differential equation">Stochastic Differential Equation</a> (SDE)

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d\mathbf {x} (t)=\mathbf {f} (t,\mathbf {x} (t))dt+\sum \limits _{i=1}^{m}\mathbf {g} _{i}(t)d\mathbf {w} ^{i}(t),\quad t\in \left[t_{0},T\right],\qquad \qquad \qquad (7.1)}</annotation>

</semantics>

</math><img src="./464be59af5611f27dd6d946030a7202ce481c427.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:70.221ex; height:6.843ex;" alt="{\displaystyle d\mathbf {x} (t)=\mathbf {f} (t,\mathbf {x} (t))dt+\sum \limits _{i=1}^{m}\mathbf {g} _{i}(t)d\mathbf {w} ^{i}(t),\quad t\in \left[t_{0},T\right],\qquad \qquad \qquad (7.1)}" loading="lazy">

</div>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} (t_{0})=\mathbf {x} _{0}}</annotation>

</semantics>

</math><img src="./4e9b78fd114617e57aff79fcd2713d7760fa660d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.678ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} (t_{0})=\mathbf {x} _{0}}" loading="lazy">, where the drift coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {f} }</annotation>

</semantics>

</math><img src="./dc6194e680a4e7c521f2178c50eea302843a852d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.053ex; height:2.176ex;" alt="{\displaystyle \mathbf {f} }" loading="lazy"> and the diffusion coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {g} _{i}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {g} _{i}}</annotation>

</semantics>

</math><img src="./e7d3fa579ce7b5d81fc8d8a05c76d81ae6dcc53c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.136ex; height:2.176ex;" alt="{\displaystyle \mathbf {g} _{i}}" loading="lazy"> are differentiable functions, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {w=(\mathbf {w} } ^{1},\ldots ,\mathbf {w} ^{m}\mathbf {)} }">

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {w=(\mathbf {w} } ^{1},\ldots ,\mathbf {w} ^{m}\mathbf {)} }</annotation>

</semantics>

</math><img src="./18c3696fa9b0429a61a5e3993ea921ac8f9a8cba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.148ex; height:3.343ex;" alt="{\displaystyle \mathbf {w=(\mathbf {w} } ^{1},\ldots ,\mathbf {w} ^{m}\mathbf {)} }" loading="lazy"> is an m-dimensional standard <a href="Wiener_process" title="Wiener process">Wiener process</a>.

<div class="mw-heading mw-heading3"><h3 id="Local_linear_discretization_4">Local linear discretization</h3></div>

<msub>

<mrow>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(t\right)_{h}}</annotation>

</semantics>

</math><img src="./16c122098f93662ae323d6fe099721b11f92825b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.828ex; height:3.009ex;" alt="{\displaystyle \left(t\right)_{h}}" loading="lazy"> , the order-<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {\gamma } }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {\gamma } }</annotation>

</semantics>

</math><img src="./51fbd8ad348bd3a53bd9626f76e09acb5c297f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \mathbb {\gamma } }" loading="lazy"> (=1,1.5) Strong Local Linear discretization of the solution of the SDE (7.1) is defined by the recursive relation <a href="#cite_note-:14-18">[18]</a><a href="#cite_note-:20-19">[19]</a>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} _{n+1}=\mathbf {z} _{n}+\mathbf {\phi } _{\mathbb {\gamma } }(t_{n},\mathbf {z} _{n};h_{n})+\mathbf {\xi } (t_{n},\mathbf {z} _{n};h_{n}),\quad with\quad \mathbf {z} _{0}=\mathbf {x} _{0},}</annotation>

</semantics>

</math><img src="./959e5fdd94446e3038e0e897c46480ef8102f1a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:59.809ex; height:3.176ex;" alt="{\displaystyle \mathbf {z} _{n+1}=\mathbf {z} _{n}+\mathbf {\phi } _{\mathbb {\gamma } }(t_{n},\mathbf {z} _{n};h_{n})+\mathbf {\xi } (t_{n},\mathbf {z} _{n};h_{n}),\quad with\quad \mathbf {z} _{0}=\mathbf {x} _{0},}" loading="lazy">

</div>

where

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\phi } _{\mathbb {\gamma } }(t_{n},\mathbf {z} _{n};\delta )=\int _{0}^{\delta }e^{\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {y} _{n})(\delta -u)}(\mathbf {f(} t_{n},\mathbf {z} _{n})+\mathbf {a} ^{\mathbb {\gamma } }(t_{n},\mathbf {z} _{n})u)du}</annotation>

</semantics>

</math><img src="./be863ef29e387aeb98831353e60505304d6af0ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:57.942ex; height:6.343ex;" alt="{\displaystyle \mathbf {\phi } _{\mathbb {\gamma } }(t_{n},\mathbf {z} _{n};\delta )=\int _{0}^{\delta }e^{\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {y} _{n})(\delta -u)}(\mathbf {f(} t_{n},\mathbf {z} _{n})+\mathbf {a} ^{\mathbb {\gamma } }(t_{n},\mathbf {z} _{n})u)du}" loading="lazy">

</div>

and

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

<msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\xi } \left(t_{n},\mathbf {z} _{n};\delta \right)=\sum \limits _{i=1}^{m}\int \nolimits _{t_{n}}^{t_{n}+\delta }e^{\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {z} _{n})(t_{n}+\delta -u)}\mathbf {g} _{i}(u)\,d\mathbf {w} ^{i}(u).}</annotation>

</semantics>

</math><img src="./7a224e4d9a7b6e97ebbe166f08941c0f078758b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:52.75ex; height:7.009ex;" alt="{\displaystyle \mathbf {\xi } \left(t_{n},\mathbf {z} _{n};\delta \right)=\sum \limits _{i=1}^{m}\int \nolimits _{t_{n}}^{t_{n}+\delta }e^{\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {z} _{n})(t_{n}+\delta -u)}\mathbf {g} _{i}(u)\,d\mathbf {w} ^{i}(u).}" loading="lazy">

</div>

Here,

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} ^{\mathbb {\gamma } }(t_{n},\mathbf {z} _{n})=\left\{{\begin{array}{cl}\mathbf {f} _{t}(t_{n},\mathbf {z} _{n})&{\text{for }}\qquad \mathbb {\gamma } =1\\\mathbf {f} _{t}(t_{n},\mathbf {z} _{n})+{\frac {1}{2}}\sum \limits _{j=1}^{m}(\mathbf {I} \otimes \mathbf {g} _{j}^{\intercal }(t_{n}))\mathbf {f} _{\mathbf {xx} }(t_{n},\mathbf {z} _{n})\mathbf {g} _{j}(t_{n})&{\text{for }}\quad \mathbb {\gamma } =1.5,\end{array}}\right.}</annotation>

</semantics>

</math><img src="./8a67cd410ed5dd2f8fef1b1aa2baa43253ce0b93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:76.732ex; height:9.843ex;" alt="{\displaystyle \mathbf {a} ^{\mathbb {\gamma } }(t_{n},\mathbf {z} _{n})=\left\{{\begin{array}{cl}\mathbf {f} _{t}(t_{n},\mathbf {z} _{n})&{\text{for }}\qquad \mathbb {\gamma } =1\\\mathbf {f} _{t}(t_{n},\mathbf {z} _{n})+{\frac {1}{2}}\sum \limits _{j=1}^{m}(\mathbf {I} \otimes \mathbf {g} _{j}^{\intercal }(t_{n}))\mathbf {f} _{\mathbf {xx} }(t_{n},\mathbf {z} _{n})\mathbf {g} _{j}(t_{n})&{\text{for }}\quad \mathbb {\gamma } =1.5,\end{array}}\right.}" loading="lazy">

</div>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {f} _{\mathbf {x} },\mathbf {f} _{t}}</annotation>

</semantics>

</math><img src="./023cb3ae4d0c62a4906f509d89f50f917dfd9a3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.723ex; height:2.509ex;" alt="{\displaystyle \mathbf {f} _{\mathbf {x} },\mathbf {f} _{t}}" loading="lazy"> denote the partial derivatives of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {f} }</annotation>

</semantics>

</math><img src="./dc6194e680a4e7c521f2178c50eea302843a852d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.053ex; height:2.176ex;" alt="{\displaystyle \mathbf {f} }" loading="lazy"> with respect to the variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation>

</semantics>

</math><img src="./32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }" loading="lazy"> and t, respectively, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} _{\mathbf {xx} }}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {f} _{\mathbf {xx} }}</annotation>

</semantics>

</math><img src="./0eaeec93fe4af91ad7b0eed791c85ecfa3c87c71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.044ex; height:2.509ex;" alt="{\displaystyle \mathbf {f} _{\mathbf {xx} }}" loading="lazy"> the Hessian matrix of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {f} }</annotation>

</semantics>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation>

</semantics>

</math><img src="./32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }" loading="lazy">. The strong Local Linear discretization <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {z} _{n+1}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} _{n+1}}</annotation>

</semantics>

</math><img src="./de9a3edf3238339d00958d22299f09e066fa2fa7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.507ex; height:2.009ex;" alt="{\displaystyle \mathbf {z} _{n+1}}" loading="lazy"> <a href="Convergence_of_random_variables" title="Convergence of random variables">converges</a> with order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {\gamma } }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {\gamma } }</annotation>

</semantics>

<div class="mw-heading mw-heading3"><h3 id="High-order_local_linear_discretizations_2">High-order local linear discretizations</h3></div>

After the local linearization of the drift term of (7.1) at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t_{n},\mathbf {z} _{n})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (t_{n},\mathbf {z} _{n})}</annotation>

</semantics>

</math><img src="./7bb5fe3b2f15bcac96ab06fb4e3a2e28f884592d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.308ex; height:2.843ex;" alt="{\displaystyle (t_{n},\mathbf {z} _{n})}" loading="lazy">, the equation for the residual <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation>

</semantics>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d\mathbf {r} (t)=\mathbf {q} _{\gamma }(t_{n},\mathbf {z} _{n};t\mathbf {,\mathbf {r} } (t))\,dt+\sum \limits _{i=1}^{m}\mathbf {g} _{i}(t)\,d\mathbf {w} ^{i}(t)\mathbf {,} \qquad \mathbf {r} (t_{n})=\mathbf {0} }</annotation>

</semantics>

</math><img src="./45617ab3c99bced9870c38682d2b6a11ff2eede2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:60.248ex; height:6.843ex;" alt="{\displaystyle d\mathbf {r} (t)=\mathbf {q} _{\gamma }(t_{n},\mathbf {z} _{n};t\mathbf {,\mathbf {r} } (t))\,dt+\sum \limits _{i=1}^{m}\mathbf {g} _{i}(t)\,d\mathbf {w} ^{i}(t)\mathbf {,} \qquad \mathbf {r} (t_{n})=\mathbf {0} }" loading="lazy">

</div>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t\in \lbrack t_{n},t_{n+1}]}</annotation>

</semantics>

</math><img src="./b15f801777044ac19f6f4e6551c3e78da54f33c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.225ex; height:2.843ex;" alt="{\displaystyle t\in \lbrack t_{n},t_{n+1}]}" loading="lazy">, where

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {q} _{\gamma }(t_{n},\mathbf {z} _{n};s\mathbf {,\xi } )=\mathbf {f} (s,\mathbf {z} _{n}+\mathbf {\phi } _{\gamma }(t_{n},\mathbf {z} _{n};s-t_{n})+\mathbf {\xi } )-\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {z} _{n})\mathbf {\phi } _{\gamma }(t_{n},\mathbf {z} _{n};s-t_{n})-\mathbf {a} ^{\gamma }(t_{n},\mathbf {z} _{n})(s-t_{n})-\mathbf {f} (t_{n},\mathbf {z} _{n}).}</annotation>

</semantics>

</math><img src="./cd66a59250c1ea111a7a9f4b0495b8f715319a01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:109.426ex; height:3.176ex;" alt="{\displaystyle \mathbf {q} _{\gamma }(t_{n},\mathbf {z} _{n};s\mathbf {,\xi } )=\mathbf {f} (s,\mathbf {z} _{n}+\mathbf {\phi } _{\gamma }(t_{n},\mathbf {z} _{n};s-t_{n})+\mathbf {\xi } )-\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {z} _{n})\mathbf {\phi } _{\gamma }(t_{n},\mathbf {z} _{n};s-t_{n})-\mathbf {a} ^{\gamma }(t_{n},\mathbf {z} _{n})(s-t_{n})-\mathbf {f} (t_{n},\mathbf {z} _{n}).}" loading="lazy">

</div>

A high-order local linear discretization of the SDE (7.1) at each point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{n+1}\in (t)_{h}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{n+1}\in (t)_{h}}</annotation>

</semantics>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} _{n+1}=\mathbf {z} _{n}+\mathbf {\phi } _{\gamma }(t_{n},\mathbf {z} _{n};h_{n})+{\widetilde {\mathbf {r} }}(t_{n},\mathbf {z} _{n};h_{n}),\qquad {\text{ with }}\qquad \mathbf {z} _{0}=\mathbf {x} _{0},}</annotation>

</semantics>

</math><img src="./a17e83deef0b1cfd1e037a0868db5dd0b47af613.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:65.624ex; height:3.176ex;" alt="{\displaystyle \mathbf {z} _{n+1}=\mathbf {z} _{n}+\mathbf {\phi } _{\gamma }(t_{n},\mathbf {z} _{n};h_{n})+{\widetilde {\mathbf {r} }}(t_{n},\mathbf {z} _{n};h_{n}),\qquad {\text{ with }}\qquad \mathbf {z} _{0}=\mathbf {x} _{0},}" loading="lazy">

</div>

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widetilde {\mathbf {r} }}}</annotation>

</semantics>

</math><img src="./9efad91385bd5fe6de0c3efac748bd910aed598b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.002ex; margin-bottom: -0.336ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\widetilde {\mathbf {r} }}}" loading="lazy"> is a strong approximation to the residual <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation>

</semantics>

</math><img src="./eca0f46511c4c986c48b254073732c0bd98ae0c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }" loading="lazy"> of order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> higher than 1.5. The strong HOLL discretization <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {z} _{n+1}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} _{n+1}}</annotation>

</semantics>

</math><img src="./de9a3edf3238339d00958d22299f09e066fa2fa7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.507ex; height:2.009ex;" alt="{\displaystyle \mathbf {z} _{n+1}}" loading="lazy"> converges with order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

<div class="mw-heading mw-heading3"><h3 id="Local_linearization_schemes_4">Local linearization schemes</h3></div>

Depending on the way of computing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\phi } _{\mathbb {\gamma } }}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\phi } _{\mathbb {\gamma } }}</annotation>

</semantics>

</math><img src="./55129d3ed93c2447e964e5777076082fec11de06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.51ex; height:3.009ex;" alt="{\displaystyle \mathbf {\phi } _{\mathbb {\gamma } }}" loading="lazy"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\xi } }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\xi } }</annotation>

</semantics>

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widetilde {\mathbf {r} }}}</annotation>

</semantics>

</math><img src="./9efad91385bd5fe6de0c3efac748bd910aed598b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.002ex; margin-bottom: -0.336ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\widetilde {\mathbf {r} }}}" loading="lazy"> different numerical schemes can be obtained. Every numerical implementation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} _{n}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{n}}</annotation>

</semantics>

</math><img src="./05275acac23ef658db9a5d0b777828444fa2096c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.629ex; height:2.176ex;" alt="{\displaystyle \mathbf {y} _{n}}" loading="lazy"> of a strong Local Linear discretization <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {z} _{n}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} _{n}}</annotation>

</semantics>

<div class="mw-heading mw-heading4"><h4 id="Order_1_SLL_schemes">Order 1 SLL schemes</h4></div>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

</semantics>

</math><img src="./cd9150237325a750fd3598b61f3599ace5a18f3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:57.648ex; height:6.843ex;" alt="{\displaystyle \mathbf {y} _{n+1}=\mathbf {y} _{n}+\mathbf {L} (\mathbf {P} _{p,q}(2^{-k_{n}}\mathbf {M} _{n}h_{n}))^{2^{k_{n}}}\mathbf {r+} \sum \limits _{i=1}^{m}\mathbf {g} _{i}(t_{n})\Delta \mathbf {w} _{n}^{i},\quad }" loading="lazy"> <a href="#cite_note-:23-21">[21]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad \qquad (7.2)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \qquad \qquad (7.2)}</annotation>

</semantics>

</math><img src="./403306f77d583df3145c76faae2a2e0e63ce7509.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.071ex; height:2.843ex;" alt="{\displaystyle \qquad \qquad (7.2)}" loading="lazy">

</div>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {M} _{n}}</annotation>

</semantics>

</math><img src="./d19961bfa42f9016b7bd444e5fa9e4f2485cc492.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.756ex; height:2.509ex;" alt="{\displaystyle \mathbf {M} _{n}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {L} }</annotation>

</semantics>

</math><img src="./6f5f750865376a1a4ae2b15a00b4ff9c75a66630.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.608ex; height:2.176ex;" alt="{\displaystyle \mathbf {L} }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation>

</semantics>

</math><img src="./eca0f46511c4c986c48b254073732c0bd98ae0c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }" loading="lazy"> are defined as in (4.6), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \mathbf {w} _{n}^{i}}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta \mathbf {w} _{n}^{i}}</annotation>

</semantics>

</math><img src="./c6ff4990cd7b0172ddbb83706215e70f3e1b7b8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.086ex; height:2.843ex;" alt="{\displaystyle \Delta \mathbf {w} _{n}^{i}}" loading="lazy"> is an <a href="Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">i.i.d.</a> zero mean <a href="Normal_distribution" title="Normal distribution">Gaussian random variable</a> with variance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h_{n}}</annotation>

</semantics>

</math><img src="./dc869ca124a4bbb264113f1b3b40d78e09c055be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.557ex; height:2.509ex;" alt="{\displaystyle h_{n}}" loading="lazy">, and p + q > 1. For large systems of SDEs,<a href="#cite_note-:23-21">[21]</a> in the above scheme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {P} _{p,q}(2^{-k_{n}}\mathbf {M} _{n}h_{n}))^{2^{k_{n}}}\mathbf {r} }">

<msub>

</mrow>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbf {P} _{p,q}(2^{-k_{n}}\mathbf {M} _{n}h_{n}))^{2^{k_{n}}}\mathbf {r} }</annotation>

</semantics>

</math><img src="./9d438f4ec877f4fc7d69fc1be7380a11eec9373e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.476ex; height:3.676ex;" alt="{\displaystyle (\mathbf {P} _{p,q}(2^{-k_{n}}\mathbf {M} _{n}h_{n}))^{2^{k_{n}}}\mathbf {r} }" loading="lazy"> is replaced by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\mathbf {k} } _{m_{n},k_{n}}^{p,q}(h_{n},\mathbf {M} _{n},\mathbf {r} )}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\mathbf {k} } _{m_{n},k_{n}}^{p,q}(h_{n},\mathbf {M} _{n},\mathbf {r} )}</annotation>

</semantics>

</math><img src="./a69d9d5a630e7b87e747f5fcac221759f51814b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:17.622ex; height:3.509ex;" alt="{\displaystyle \mathbf {\mathbf {k} } _{m_{n},k_{n}}^{p,q}(h_{n},\mathbf {M} _{n},\mathbf {r} )}" loading="lazy">.

<div class="mw-heading mw-heading4"><h4 id="Order_1.5_SLL_schemes">Order 1.5 SLL schemes</h4></div>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</munderover>

<mrow>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{n+1}=\mathbf {y} _{n}+\mathbf {L} (\mathbf {P} _{p,q}(2^{-k_{n}}\mathbf {M} _{n}h_{n}))^{2^{k_{n}}}\mathbf {r} +\sum \limits _{i=1}^{m}\left(\mathbf {g} _{i}(t_{n})\Delta \mathbf {w} _{n}^{i}\mathbf {f} _{\mathbf {x} }(t_{n},{\widetilde {\mathbf {y} }}_{n})\mathbf {g} _{i}(t_{n})\Delta \mathbf {z} _{n}^{i}+{\frac {d\mathbf {g} _{i}(t_{n})}{dt}}(\Delta \mathbf {w} _{n}^{i}h_{n}-\Delta \mathbf {z} _{n}^{i})\right),\qquad \qquad (7.3)}</annotation>

</semantics>

</math><img src="./dc5d5cac78017f7c6481df96cde2b4e882bcb184.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:121.422ex; height:6.843ex;" alt="{\displaystyle \mathbf {y} _{n+1}=\mathbf {y} _{n}+\mathbf {L} (\mathbf {P} _{p,q}(2^{-k_{n}}\mathbf {M} _{n}h_{n}))^{2^{k_{n}}}\mathbf {r} +\sum \limits _{i=1}^{m}\left(\mathbf {g} _{i}(t_{n})\Delta \mathbf {w} _{n}^{i}\mathbf {f} _{\mathbf {x} }(t_{n},{\widetilde {\mathbf {y} }}_{n})\mathbf {g} _{i}(t_{n})\Delta \mathbf {z} _{n}^{i}+{\frac {d\mathbf {g} _{i}(t_{n})}{dt}}(\Delta \mathbf {w} _{n}^{i}h_{n}-\Delta \mathbf {z} _{n}^{i})\right),\qquad \qquad (7.3)}" loading="lazy">

</div>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {M} _{n}}</annotation>

</semantics>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {L} }</annotation>

</semantics>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation>

</semantics>

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {M} _{n}={\begin{bmatrix}\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {y} _{n})&\mathbf {f} _{t}(t_{n},\mathbf {y} _{n})+{\frac {1}{2}}\sum \limits _{j=1}^{m}\left(\mathbf {I} \otimes \mathbf {g} _{j}^{\intercal }(t_{n})\right)\mathbf {f} _{\mathbf {xx} }(t_{n},\mathbf {y} _{n})\mathbf {g} _{j}(t_{n})&\mathbf {f} (t_{n},\mathbf {y} _{n})\\0&0&1\\0&0&0\end{bmatrix}}\in \mathbb {R} ^{(d+2)\times (d+2)},}</annotation>

</semantics>

</math><img src="./5c6cb8b42d5c64673dbe5bfd4b9233af8eceb377.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:95.948ex; height:12.843ex;" alt="{\displaystyle \mathbf {M} _{n}={\begin{bmatrix}\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {y} _{n})&\mathbf {f} _{t}(t_{n},\mathbf {y} _{n})+{\frac {1}{2}}\sum \limits _{j=1}^{m}\left(\mathbf {I} \otimes \mathbf {g} _{j}^{\intercal }(t_{n})\right)\mathbf {f} _{\mathbf {xx} }(t_{n},\mathbf {y} _{n})\mathbf {g} _{j}(t_{n})&\mathbf {f} (t_{n},\mathbf {y} _{n})\\0&0&1\\0&0&0\end{bmatrix}}\in \mathbb {R} ^{(d+2)\times (d+2)},}" loading="lazy">

</div>

</mrow>

<mrow>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {L} =\left[{\begin{array}{ll}\mathbf {I} &\mathbf {0} _{d\times 2}\end{array}}\right],\mathbf {r} ^{\intercal }=\left[{\begin{array}{ll}\mathbf {0} _{1\times (d+1)}&1\end{array}}\right]}</annotation>

</semantics>

</math><img src="./4096111183a23671fc2d181a96e16ca7b7df6d6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.47ex; height:3.176ex;" alt="{\displaystyle \mathbf {L} =\left[{\begin{array}{ll}\mathbf {I} &\mathbf {0} _{d\times 2}\end{array}}\right],\mathbf {r} ^{\intercal }=\left[{\begin{array}{ll}\mathbf {0} _{1\times (d+1)}&1\end{array}}\right]}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \mathbf {z} _{n}^{i}}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta \mathbf {z} _{n}^{i}}</annotation>

</semantics>

</math><img src="./0ec6c63542ce6b2d7319efa03a71a0d85c59f3df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.342ex; height:2.843ex;" alt="{\displaystyle \Delta \mathbf {z} _{n}^{i}}" loading="lazy"> is a i.i.d. zero mean Gaussian random variable with variance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\left((\Delta \mathbf {z} _{n}^{i})^{2}\right)={\frac {1}{3}}h_{n}^{3}}">

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E\left((\Delta \mathbf {z} _{n}^{i})^{2}\right)={\frac {1}{3}}h_{n}^{3}}</annotation>

</semantics>

</math><img src="./ca91052a12dc974f73d9f766ba83e3f63079eb28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.153ex; height:5.176ex;" alt="{\displaystyle E\left((\Delta \mathbf {z} _{n}^{i})^{2}\right)={\frac {1}{3}}h_{n}^{3}}" loading="lazy"> and covariance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(\Delta \mathbf {w} _{n}^{i}\Delta \mathbf {z} _{n}^{i})={\frac {1}{2}}h_{n}^{2}}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E(\Delta \mathbf {w} _{n}^{i}\Delta \mathbf {z} _{n}^{i})={\frac {1}{2}}h_{n}^{2}}</annotation>

</semantics>

</math><img src="./740f22615f877ce5268cdcb46991599a0d781104.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.667ex; height:5.176ex;" alt="{\displaystyle E(\Delta \mathbf {w} _{n}^{i}\Delta \mathbf {z} _{n}^{i})={\frac {1}{2}}h_{n}^{2}}" loading="lazy"> and p+q>1 <a href="#cite_note-:12-12">[12]</a>. For large systems of SDEs,<a href="#cite_note-:12-12">[12]</a> in the above scheme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {P} _{p,q}(2^{-k_{n}}\mathbf {M} _{n}h_{n}))^{2^{k_{n}}}\mathbf {r} }">

<msub>

</mrow>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbf {P} _{p,q}(2^{-k_{n}}\mathbf {M} _{n}h_{n}))^{2^{k_{n}}}\mathbf {r} }</annotation>

</semantics>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\mathbf {k} } _{m_{n},k_{n}}^{p,q}(h_{n},\mathbf {M} _{n},\mathbf {r} )}</annotation>

</semantics>

<div class="mw-heading mw-heading4"><h4 id="Order_2_SLL-Taylor_schemes">Order 2 SLL-Taylor schemes</h4></div>

<msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

<mover>

</mover>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

<mover>

</mover>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{t_{n+1}}=\mathbf {y} _{n}+\mathbf {L} (\mathbf {P} _{p,q}(2^{-k_{n}}\mathbf {M} _{n}h_{n}))^{2^{k_{n}}}\mathbf {r} +\sum \limits _{j=1}^{m}\mathbf {g} _{j}\left(t_{n}\right)\Delta \mathbf {w} _{n}^{j}+\sum \limits _{j=1}^{m}\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {y} _{n})\mathbf {g} _{j}\left(t_{n}\right){\widetilde {J}}_{\left(j,0\right)}+\sum \limits _{j=1}^{m}{\frac {d\mathbf {g} _{_{j}}}{dt}}\left(t_{n}\right){\widetilde {J}}_{\left(0,j\right)}}</annotation>

</semantics>

</math><img src="./262f0bfabce6ad7da6be4fd176fddc894f3c4a79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:105.435ex; height:7.509ex;" alt="{\displaystyle \mathbf {y} _{t_{n+1}}=\mathbf {y} _{n}+\mathbf {L} (\mathbf {P} _{p,q}(2^{-k_{n}}\mathbf {M} _{n}h_{n}))^{2^{k_{n}}}\mathbf {r} +\sum \limits _{j=1}^{m}\mathbf {g} _{j}\left(t_{n}\right)\Delta \mathbf {w} _{n}^{j}+\sum \limits _{j=1}^{m}\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {y} _{n})\mathbf {g} _{j}\left(t_{n}\right){\widetilde {J}}_{\left(j,0\right)}+\sum \limits _{j=1}^{m}{\frac {d\mathbf {g} _{_{j}}}{dt}}\left(t_{n}\right){\widetilde {J}}_{\left(0,j\right)}}" loading="lazy">

</div>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<mrow>

<mrow>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

<mover>

</mover>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \qquad \qquad +\sum \limits _{j_{1},j_{2}=1}^{m}\left(\mathbf {I} \otimes \mathbf {g} _{j_{2}}^{\intercal }\left(t_{n}\right)\right)\mathbf {f} _{\mathbf {xx} }(t_{n},\mathbf {y} _{n})\mathbf {g} _{j_{1}}\left(t_{n}\right){\widetilde {J}}_{\left(j_{1},j_{2},0\right),}\qquad \qquad (7.4)}</annotation>

</semantics>

</math><img src="./f8dd5a73de68a6d4d4869bf3f4ad642a209f8783.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:72.775ex; height:7.343ex;" alt="{\displaystyle \qquad \qquad +\sum \limits _{j_{1},j_{2}=1}^{m}\left(\mathbf {I} \otimes \mathbf {g} _{j_{2}}^{\intercal }\left(t_{n}\right)\right)\mathbf {f} _{\mathbf {xx} }(t_{n},\mathbf {y} _{n})\mathbf {g} _{j_{1}}\left(t_{n}\right){\widetilde {J}}_{\left(j_{1},j_{2},0\right),}\qquad \qquad (7.4)}" loading="lazy">

</div>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {M} _{n}}</annotation>

</semantics>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {L} }</annotation>

</semantics>

</math><img src="./6f5f750865376a1a4ae2b15a00b4ff9c75a66630.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.608ex; height:2.176ex;" alt="{\displaystyle \mathbf {L} }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation>

</semantics>

</math><img src="./eca0f46511c4c986c48b254073732c0bd98ae0c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \mathbf {w} _{n}^{i}}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta \mathbf {w} _{n}^{i}}</annotation>

</semantics>

</math><img src="./c6ff4990cd7b0172ddbb83706215e70f3e1b7b8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.086ex; height:2.843ex;" alt="{\displaystyle \Delta \mathbf {w} _{n}^{i}}" loading="lazy"> are defined as in the order-1 SLL schemes, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {J}}_{\alpha }}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widetilde {J}}_{\alpha }}</annotation>

</semantics>

</math><img src="./6fd48eae88314c1a3bcf1f5fe27d4e1a3d1e71e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.172ex; height:3.009ex;" alt="{\displaystyle {\widetilde {J}}_{\alpha }}" loading="lazy"> is order 2 approximation to the multiple <a href="Stratonovich_integral" title="Stratonovich integral">Stratonovish integral</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{\alpha }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J_{\alpha }}</annotation>

</semantics>

</math><img src="./12aec95e5551460fd54f17ebcf44ffb6e6d3e7f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.574ex; height:2.509ex;" alt="{\displaystyle J_{\alpha }}" loading="lazy">.<a href="#cite_note-:4-20">[20]</a>

<div class="mw-heading mw-heading4"><h4 id="Order_2_SLL-RK_schemes">Order 2 SLL-RK schemes</h4></div>

For SDEs with a single Wiener noise (m=1) <a href="#cite_note-:4-20">[20]</a>

<msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<msub>

<mrow>

<mrow>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{t_{n+1}}=\mathbf {y} _{n}+{\widetilde {\mathbf {\phi } }}(t_{n},\mathbf {y} _{n};h_{n})+{\frac {h_{n}}{2}}\left(\mathbf {k} _{1}+\mathbf {k} _{2}\right)+\mathbf {g} \left(t_{n}\right)\Delta w_{n}+{\frac {\left(\mathbf {g} \left(t_{n+1}\right)-\mathbf {g} \left(t_{n}\right)\right)}{h_{n}}}J_{\left(0,1\right)}\quad (7.5)}</annotation>

</semantics>

</math><img src="./44925c284686a48f4909bde8e43923c9cefa738b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:89.584ex; height:6.176ex;" alt="{\displaystyle \mathbf {y} _{t_{n+1}}=\mathbf {y} _{n}+{\widetilde {\mathbf {\phi } }}(t_{n},\mathbf {y} _{n};h_{n})+{\frac {h_{n}}{2}}\left(\mathbf {k} _{1}+\mathbf {k} _{2}\right)+\mathbf {g} \left(t_{n}\right)\Delta w_{n}+{\frac {\left(\mathbf {g} \left(t_{n+1}\right)-\mathbf {g} \left(t_{n}\right)\right)}{h_{n}}}J_{\left(0,1\right)}\quad (7.5)}" loading="lazy">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \quad \quad \quad }</annotation>

</semantics>

</math><img src="./b64ae17666c2046aa2fe98cd78b05da6358f4a2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:6.968ex; height:0.343ex;" alt="{\displaystyle \quad \quad \quad }" loading="lazy">

where

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {k} _{1}=\mathbf {f} (t_{n}+{\frac {h_{n}}{2}},\mathbf {y} _{n}+{\widetilde {\mathbf {\phi } }}(t_{n},\mathbf {y} _{n};{\frac {h_{n}}{2}})+\gamma _{+})-\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {y} _{n}){\widetilde {\mathbf {\phi } }}(t_{n},\mathbf {y} _{n};{\frac {h_{n}}{2}})-\mathbf {f} \left(t_{n},\mathbf {y} _{n}\right)-\mathbf {f} _{t}\left(t_{n},\mathbf {y} _{n}\right){\frac {h_{n}}{2}},}</annotation>

</semantics>

</math><img src="./bc3239f0408181fae310b575deb4564460cfd43d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:96.816ex; height:5.343ex;" alt="{\displaystyle \mathbf {k} _{1}=\mathbf {f} (t_{n}+{\frac {h_{n}}{2}},\mathbf {y} _{n}+{\widetilde {\mathbf {\phi } }}(t_{n},\mathbf {y} _{n};{\frac {h_{n}}{2}})+\gamma _{+})-\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {y} _{n}){\widetilde {\mathbf {\phi } }}(t_{n},\mathbf {y} _{n};{\frac {h_{n}}{2}})-\mathbf {f} \left(t_{n},\mathbf {y} _{n}\right)-\mathbf {f} _{t}\left(t_{n},\mathbf {y} _{n}\right){\frac {h_{n}}{2}},}" loading="lazy"></dd></dl>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {k} _{2}=\mathbf {f} (t_{n}+{\frac {h_{n}}{2}},\mathbf {y} _{n}+{\widetilde {\mathbf {\phi } }}(t_{n},\mathbf {y} _{n};{\frac {h_{n}}{2}})+\gamma _{-})-\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {y} _{n}){\widetilde {\mathbf {\phi } }}(t_{n},\mathbf {y} _{n};{\frac {h_{n}}{2}})-\mathbf {f} \left(t_{n},\mathbf {y} _{n}\right)-\mathbf {f} _{t}\left(t_{n},\mathbf {y} _{n}\right){\frac {h_{n}}{2}},}</annotation>

</semantics>

</math><img src="./a9eb573b535abdfe605611ebf1e93ee26d3df836.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:96.816ex; height:5.343ex;" alt="{\displaystyle \mathbf {k} _{2}=\mathbf {f} (t_{n}+{\frac {h_{n}}{2}},\mathbf {y} _{n}+{\widetilde {\mathbf {\phi } }}(t_{n},\mathbf {y} _{n};{\frac {h_{n}}{2}})+\gamma _{-})-\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {y} _{n}){\widetilde {\mathbf {\phi } }}(t_{n},\mathbf {y} _{n};{\frac {h_{n}}{2}})-\mathbf {f} \left(t_{n},\mathbf {y} _{n}\right)-\mathbf {f} _{t}\left(t_{n},\mathbf {y} _{n}\right){\frac {h_{n}}{2}},}" loading="lazy"></dd></dl>

with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{\pm }={\frac {1}{h_{n}}}\mathbf {g} \left(t_{n}\right){\Bigl (}{\widetilde {J}}_{\left(1,0\right)}\pm {\sqrt {2{\widetilde {J}}_{\left(1,1,0\right)}h_{n}-{\widetilde {J}}_{\left(1,0\right)}^{2}}}{\Bigr )}}">

<msub>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

<mover>

</mover>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</msub>

<msqrt>

<msub>

<mover>

</mover>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma _{\pm }={\frac {1}{h_{n}}}\mathbf {g} \left(t_{n}\right){\Bigl (}{\widetilde {J}}_{\left(1,0\right)}\pm {\sqrt {2{\widetilde {J}}_{\left(1,1,0\right)}h_{n}-{\widetilde {J}}_{\left(1,0\right)}^{2}}}{\Bigr )}}</annotation>

</semantics>

</math><img src="./7aaf40a5d61fa220c087aa7aa0c7ba7facf4c70c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:47.466ex; height:5.676ex;" alt="{\displaystyle \gamma _{\pm }={\frac {1}{h_{n}}}\mathbf {g} \left(t_{n}\right){\Bigl (}{\widetilde {J}}_{\left(1,0\right)}\pm {\sqrt {2{\widetilde {J}}_{\left(1,1,0\right)}h_{n}-{\widetilde {J}}_{\left(1,0\right)}^{2}}}{\Bigr )}}" loading="lazy">.

Here, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {\mathbf {\phi } }}(t_{n},\mathbf {y} _{n};h_{n})=\mathbf {L} (\mathbf {P} _{p,q}(2^{-k_{n}}\mathbf {M} _{n}h_{n}))^{2^{k_{n}}}\mathbf {r} }">

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widetilde {\mathbf {\phi } }}(t_{n},\mathbf {y} _{n};h_{n})=\mathbf {L} (\mathbf {P} _{p,q}(2^{-k_{n}}\mathbf {M} _{n}h_{n}))^{2^{k_{n}}}\mathbf {r} }</annotation>

</semantics>

</math><img src="./db4ab4c67f3836b770fcc448d7e3a86f7cac64e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.837ex; height:3.676ex;" alt="{\displaystyle {\widetilde {\mathbf {\phi } }}(t_{n},\mathbf {y} _{n};h_{n})=\mathbf {L} (\mathbf {P} _{p,q}(2^{-k_{n}}\mathbf {M} _{n}h_{n}))^{2^{k_{n}}}\mathbf {r} }" loading="lazy"> for low dimensional SDEs, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {\mathbf {\phi } }}(t_{n},\mathbf {y} _{n};h_{n})=\mathbf {L\mathbf {k} } _{m_{n},k_{n}}^{p,q}(h_{n},\mathbf {M} _{n},\mathbf {r} )}">

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widetilde {\mathbf {\phi } }}(t_{n},\mathbf {y} _{n};h_{n})=\mathbf {L\mathbf {k} } _{m_{n},k_{n}}^{p,q}(h_{n},\mathbf {M} _{n},\mathbf {r} )}</annotation>

</semantics>

</math><img src="./c02720a13d0c2d1df36d44d18671afb76352e2ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:34.984ex; height:3.676ex;" alt="{\displaystyle {\widetilde {\mathbf {\phi } }}(t_{n},\mathbf {y} _{n};h_{n})=\mathbf {L\mathbf {k} } _{m_{n},k_{n}}^{p,q}(h_{n},\mathbf {M} _{n},\mathbf {r} )}" loading="lazy"> for large systems of SDEs, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} _{n}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {M} _{n}}</annotation>

</semantics>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {L} }</annotation>

</semantics>

</math><img src="./6f5f750865376a1a4ae2b15a00b4ff9c75a66630.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.608ex; height:2.176ex;" alt="{\displaystyle \mathbf {L} }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation>

</semantics>

</math><img src="./eca0f46511c4c986c48b254073732c0bd98ae0c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \mathbf {w} _{n}^{i}}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta \mathbf {w} _{n}^{i}}</annotation>

</semantics>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widetilde {J}}_{\alpha }}</annotation>

</semantics>

</math><img src="./6fd48eae88314c1a3bcf1f5fe27d4e1a3d1e71e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.172ex; height:3.009ex;" alt="{\displaystyle {\widetilde {J}}_{\alpha }}" loading="lazy"> are defined as in the order-2 SLL-Taylor schemes, p+q>1 and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{n}>2}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m_{n}>2}</annotation>

</semantics>

<div class="mw-heading mw-heading4"><h4 id="Stability_and_dynamics_2">Stability and dynamics</h4></div>

By construction, the strong LL and HOLL discretizations inherit the stability and <a href="Random_dynamical_system" title="Random dynamical system">dynamics</a> of the linear SDEs, but it is not the case of the strong LL schemes in general. LL schemes (7.2)-(7.5) with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\leq q\leq p+2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\leq q\leq p+2}</annotation>

</semantics>

</math><img src="./fdc7e589b60294374f26e615dbcba5d137684a65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:13.698ex; height:2.509ex;" alt="{\displaystyle p\leq q\leq p+2}" loading="lazy"> are A-stable, including stiff and highly oscillatory linear equations.<a href="#cite_note-:12-12">[12]</a> Moreover, for linear SDEs with <a href="Pullback_attractor" title="Pullback attractor">random attractors</a>, these schemes also have a random attractor that <a href="Convergence_in_probability" class="mw-redirect" title="Convergence in probability">converges in probability</a> to the exact one as the stepsize decreases and preserve the <a href="Ergodicity" title="Ergodicity">ergodicity</a> of these equations for any stepsize.<a href="#cite_note-:4-20">[20]</a><a href="#cite_note-:12-12">[12]</a> These schemes also reproduce essential dynamical properties of simple and coupled harmonic oscillators such as the linear growth of energy along the paths, the oscillatory behavior around 0, the symplectic structure of Hamiltonian oscillators, and the mean of the paths.<a href="#cite_note-:4-20">[20]</a><a href="#cite_note-:5-22">[22]</a> For nonlinear SDEs with small noise (i.e., (7.1) with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {g} _{i}(t)\approx 0}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {g} _{i}(t)\approx 0}</annotation>

</semantics>

</math><img src="./d413c2850a8e0526f4d32dfa70782308c627f3da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.046ex; height:2.843ex;" alt="{\displaystyle \mathbf {g} _{i}(t)\approx 0}" loading="lazy">), the paths of these SLL schemes are basically the nonrandom paths of the LL scheme (4.6) for ODEs plus a small disturbance related to the small noise. In this situation, the dynamical properties of that deterministic scheme, such as the linearization preserving and the preservation of the exact solution dynamics around hyperbolic equilibrium points and periodic orbits, become relevant for the paths of the SLL scheme.<a href="#cite_note-:4-20">[20]</a> For instance, Fig 4 shows the evolution of domains in the phase plane and the energy of the stochastic oscillator

<mtr>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{ll}dx(t)=y(t)dt,&x_{1}(0)=0.01\\dy(t)=-(\omega ^{2}x(t)+\epsilon x^{4}(t))dt+\sigma dw_{t},&x_{1}(0)=0.1,\end{array}}\qquad \qquad (7.6)}</annotation>

</semantics>

</math><img src="./bdc8edc8c3652d3b9a48a1029cbae29ebc82ae9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.374ex; margin-bottom: -0.297ex; width:67.344ex; height:6.509ex;" alt="{\displaystyle {\begin{array}{ll}dx(t)=y(t)dt,&x_{1}(0)=0.01\\dy(t)=-(\omega ^{2}x(t)+\epsilon x^{4}(t))dt+\sigma dw_{t},&x_{1}(0)=0.1,\end{array}}\qquad \qquad (7.6)}" loading="lazy">

and their approximations by two numerical schemes.

<div class="mw-heading mw-heading2"><h2 id="Weak_LL_methods_for_SDEs">Weak LL methods for SDEs</h2></div>

Consider the d-dimensional stochastic differential equation

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d\mathbf {x} (t)=\mathbf {f} (t,\mathbf {x} (t))dt+\sum \limits _{i=1}^{m}\mathbf {g} _{i}(t)d\mathbf {w} ^{i}(t),\qquad t\in \left[t_{0},T\right],\qquad \qquad (8.1)}</annotation>

</semantics>

</math><img src="./4db0bd45e90a7c91e2715a14caf58e220b2e0b36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:67.899ex; height:6.843ex;" alt="{\displaystyle d\mathbf {x} (t)=\mathbf {f} (t,\mathbf {x} (t))dt+\sum \limits _{i=1}^{m}\mathbf {g} _{i}(t)d\mathbf {w} ^{i}(t),\qquad t\in \left[t_{0},T\right],\qquad \qquad (8.1)}" loading="lazy">

</div>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} (t_{0})=\mathbf {x} _{0}}</annotation>

</semantics>

</math><img src="./4e9b78fd114617e57aff79fcd2713d7760fa660d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.678ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} (t_{0})=\mathbf {x} _{0}}" loading="lazy">, where the drift coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {f} }</annotation>

</semantics>

</math><img src="./dc6194e680a4e7c521f2178c50eea302843a852d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.053ex; height:2.176ex;" alt="{\displaystyle \mathbf {f} }" loading="lazy"> and the diffusion coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {g} _{i}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {g} _{i}}</annotation>

</semantics>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {w=(\mathbf {w} } ^{1},\ldots ,\mathbf {w} ^{m}\mathbf {)} }</annotation>

</semantics>

<div class="mw-heading mw-heading3"><h3 id="Local_Linear_discretization_5">Local Linear discretization</h3></div>

<msub>

<mrow>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(t\right)_{h}}</annotation>

</semantics>

</math><img src="./16c122098f93662ae323d6fe099721b11f92825b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.828ex; height:3.009ex;" alt="{\displaystyle \left(t\right)_{h}}" loading="lazy">, the order-<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {\beta } }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {\beta } }</annotation>

</semantics>

</math><img src="./179baa511cbe529272df4adf53f7e2724c30e763.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \mathbb {\beta } }" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (=1,2)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (=1,2)}</annotation>

</semantics>

</math><img src="./6a9d1ab408f8cd7cd931c2be081a456985e2f66b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.621ex; height:2.843ex;" alt="{\displaystyle (=1,2)}" loading="lazy"> Weak Local Linear discretization of the solution of the SDE (8.1) is defined by the recursive relation <a href="#cite_note-:21-23">[23]</a>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} _{n+1}=\mathbf {z} _{n}+\mathbf {\phi } _{\mathbb {\beta } }(t_{n},\mathbf {z} _{n};h_{n})+\mathbf {\eta } (t_{n},\mathbf {z} _{n};h_{n}),\quad with\quad \mathbf {z} _{0}=\mathbf {x} _{0},}</annotation>

</semantics>

</math><img src="./19297a288ac1d09b1bbca6d18da73ee65ddd045e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:59.997ex; height:3.176ex;" alt="{\displaystyle \mathbf {z} _{n+1}=\mathbf {z} _{n}+\mathbf {\phi } _{\mathbb {\beta } }(t_{n},\mathbf {z} _{n};h_{n})+\mathbf {\eta } (t_{n},\mathbf {z} _{n};h_{n}),\quad with\quad \mathbf {z} _{0}=\mathbf {x} _{0},}" loading="lazy">

</div>

where

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\phi } _{\mathbb {\beta } }(t_{n},\mathbf {z} _{n};\delta )=\int _{0}^{\delta }e^{\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {z} _{n})(\delta -u)}(\mathbf {f(} t_{n},\mathbf {z} _{n})+\mathbf {b} ^{\mathbb {\beta } }(t_{n},\mathbf {z} _{n})u)du}</annotation>

</semantics>

</math><img src="./d99ec267bbc944048e56342f21c1a25a3d71c7f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:58.069ex; height:6.343ex;" alt="{\displaystyle \mathbf {\phi } _{\mathbb {\beta } }(t_{n},\mathbf {z} _{n};\delta )=\int _{0}^{\delta }e^{\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {z} _{n})(\delta -u)}(\mathbf {f(} t_{n},\mathbf {z} _{n})+\mathbf {b} ^{\mathbb {\beta } }(t_{n},\mathbf {z} _{n})u)du}" loading="lazy">

</div>

with

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {b} ^{\mathbb {\beta } }(t_{n},\mathbf {z} _{n})={\begin{cases}\mathbf {f} _{t}(t_{n},\mathbf {z} _{n})&{\text{for }}\mathbb {\beta } =1\\\mathbf {f} _{t}(t_{n},\mathbf {z} _{n})+{\frac {1}{2}}\sum \limits _{j=1}^{m}\left(\mathbf {I} \otimes \mathbf {g} _{j}^{\intercal }\left(t_{n}\right)\right)\mathbf {f} _{\mathbf {xx} }(t_{n},\mathbf {z} _{n})\mathbf {g} _{j}\left(t_{n}\right)&{\text{for }}\mathbb {\beta } =2,\end{cases}}}</annotation>

</semantics>

</math><img src="./ce3ccf884eaad396acc2281b25d379ba2375c97b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:75.42ex; height:9.176ex;" alt="{\displaystyle \mathbf {b} ^{\mathbb {\beta } }(t_{n},\mathbf {z} _{n})={\begin{cases}\mathbf {f} _{t}(t_{n},\mathbf {z} _{n})&{\text{for }}\mathbb {\beta } =1\\\mathbf {f} _{t}(t_{n},\mathbf {z} _{n})+{\frac {1}{2}}\sum \limits _{j=1}^{m}\left(\mathbf {I} \otimes \mathbf {g} _{j}^{\intercal }\left(t_{n}\right)\right)\mathbf {f} _{\mathbf {xx} }(t_{n},\mathbf {z} _{n})\mathbf {g} _{j}\left(t_{n}\right)&{\text{for }}\mathbb {\beta } =2,\end{cases}}}" loading="lazy">

</div>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\eta } (t_{n},\mathbf {z} _{n};\delta )}</annotation>

</semantics>

</math><img src="./6d871c72a497e8e95e15b3d4eee3ce242f67728e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.56ex; height:2.843ex;" alt="{\displaystyle \mathbf {\eta } (t_{n},\mathbf {z} _{n};\delta )}" loading="lazy"> is a zero mean stochastic process with variance matrix

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\Sigma } (t_{n},\mathbf {z} _{n};\delta )=\int \limits _{0}^{\delta }e^{\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {z} _{n})(\delta -s)}\mathbf {G} (t_{n}+s)\mathbf {G} ^{\intercal }(t_{n}+s)e^{\mathbf {f} _{\mathbf {x} }^{\intercal }(t_{n},\mathbf {z} _{n})(\delta -s)}ds.}</annotation>

</semantics>

</math><img src="./37c5f26d8987e4b76cd639b521900dba595a5774.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:65.124ex; height:9.343ex;" alt="{\displaystyle \mathbf {\Sigma } (t_{n},\mathbf {z} _{n};\delta )=\int \limits _{0}^{\delta }e^{\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {z} _{n})(\delta -s)}\mathbf {G} (t_{n}+s)\mathbf {G} ^{\intercal }(t_{n}+s)e^{\mathbf {f} _{\mathbf {x} }^{\intercal }(t_{n},\mathbf {z} _{n})(\delta -s)}ds.}" loading="lazy">

</div>

Here, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} _{\mathbf {x} }}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {f} _{\mathbf {x} }}</annotation>

</semantics>

</math><img src="./0d1fff5ced74f815fed8a4b4f1f66b154f7e0ccb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.046ex; height:2.509ex;" alt="{\displaystyle \mathbf {f} _{\mathbf {x} }}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} _{t}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {f} _{t}}</annotation>

</semantics>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {f} }</annotation>

</semantics>

</math><img src="./dc6194e680a4e7c521f2178c50eea302843a852d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.053ex; height:2.176ex;" alt="{\displaystyle \mathbf {f} }" loading="lazy"> with respect to the variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation>

</semantics>

</math><img src="./32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }" loading="lazy"> and t, respectively, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} _{\mathbf {xx} }}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {f} _{\mathbf {xx} }}</annotation>

</semantics>

</math><img src="./0eaeec93fe4af91ad7b0eed791c85ecfa3c87c71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.044ex; height:2.509ex;" alt="{\displaystyle \mathbf {f} _{\mathbf {xx} }}" loading="lazy"> the Hessian matrix of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {f} }</annotation>

</semantics>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation>

</semantics>

</math><img src="./32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {G} (t)=[\mathbf {g} _{1}(t),\ldots ,\mathbf {g} _{m}(t)]}">

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {G} (t)=[\mathbf {g} _{1}(t),\ldots ,\mathbf {g} _{m}(t)]}</annotation>

</semantics>

</math><img src="./12433070ae8deb0a55bf46601f3b7c2b4c410c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.02ex; height:2.843ex;" alt="{\displaystyle \mathbf {G} (t)=[\mathbf {g} _{1}(t),\ldots ,\mathbf {g} _{m}(t)]}" loading="lazy">. The weak Local Linear discretization <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {z} _{n+1}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} _{n+1}}</annotation>

</semantics>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {\beta } }</annotation>

</semantics>

<div class="mw-heading mw-heading3"><h3 id="Local_Linearization_schemes_5">Local Linearization schemes</h3></div>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\phi } _{\mathbb {\beta } }}</annotation>

</semantics>

</math><img src="./c91ca5e9884c1c60d4b480b1487eda4057a8e0e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.56ex; height:3.009ex;" alt="{\displaystyle \mathbf {\phi } _{\mathbb {\beta } }}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\Sigma } }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\Sigma } }</annotation>

</semantics>

</math><img src="./90f99b56fe6ada781ecd0f8a45b6e787b6dfed56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle \mathbf {\Sigma } }" loading="lazy"> different numerical schemes can be obtained. Every numerical implementation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} _{n}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{n}}</annotation>

</semantics>

</math><img src="./05275acac23ef658db9a5d0b777828444fa2096c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.629ex; height:2.176ex;" alt="{\displaystyle \mathbf {y} _{n}}" loading="lazy"> of the Weak Local Linear discretization <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {z} _{n}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} _{n}}</annotation>

</semantics>

<div class="mw-heading mw-heading4"><h4 id="Order_1_WLL_scheme">Order 1 WLL scheme</h4></div>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{n+1}=\mathbf {y} _{n}+\mathbf {B} _{14}+(\mathbf {B} _{12}\mathbf {B} _{11}^{\intercal })^{1/2}\mathbf {\xi } _{n}}</annotation>

</semantics>

</math><img src="./024b8e88186566498ddf487265aa1b57dc7e47b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.215ex; height:3.509ex;" alt="{\displaystyle \mathbf {y} _{n+1}=\mathbf {y} _{n}+\mathbf {B} _{14}+(\mathbf {B} _{12}\mathbf {B} _{11}^{\intercal })^{1/2}\mathbf {\xi } _{n}}" loading="lazy">

</div>

where, for SDEs with autonomous diffusion coefficients, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} _{11}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {B} _{11}}</annotation>

</semantics>

</math><img src="./a2a3c364a3804ba0daccd0d08690555485f043a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.777ex; height:2.509ex;" alt="{\displaystyle \mathbf {B} _{11}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} _{12}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {B} _{12}}</annotation>

</semantics>

</math><img src="./b1d2b6bb7e128b358fee48e01651a40ef3c0c704.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.777ex; height:2.509ex;" alt="{\displaystyle \mathbf {B} _{12}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} _{14}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {B} _{14}}</annotation>

</semantics>

</math><img src="./7a7a7a9c63a875256955242ec6f03b5018f6e581.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.777ex; height:2.509ex;" alt="{\displaystyle \mathbf {B} _{14}}" loading="lazy"> are the submatrices defined by the <a href="Block_matrix" title="Block matrix">partitioned matrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} =\mathbf {P} _{p,q}(2^{-k_{n}}{\mathcal {M}}_{n}h_{n}))^{2^{k_{n}}}}">

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {B} =\mathbf {P} _{p,q}(2^{-k_{n}}{\mathcal {M}}_{n}h_{n}))^{2^{k_{n}}}}</annotation>

</semantics>

</math><img src="./16a8742182255e2b453f27366aa96d1ca5e57e94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.722ex; height:3.676ex;" alt="{\displaystyle \mathbf {B} =\mathbf {P} _{p,q}(2^{-k_{n}}{\mathcal {M}}_{n}h_{n}))^{2^{k_{n}}}}" loading="lazy">, with

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msup>

</mrow>

</mrow>

</msup>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}_{n}=\left[{\begin{array}{cccc}\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {y} _{n})&\mathbf {GG} ^{\intercal }&\mathbf {f} _{t}(t_{n},\mathbf {y} _{n})&\mathbf {f} (t_{n},\mathbf {y} _{n})\\\mathbf {0} &-\mathbf {f} _{\mathbf {x} }^{\intercal }(t_{n},\mathbf {y} _{n})&\mathbf {0} &\mathbf {0} \\\mathbf {0} &\mathbf {0} &0&1\\\mathbf {0} &\mathbf {0} &0&0\end{array}}\right]\in \mathbb {R} ^{(2d+2)\times (2d+2)},}</annotation>

</semantics>

</math><img src="./62b6c5cb6cc2c1a036002082a4dcad32b5080660.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:73.671ex; height:12.843ex;" alt="{\displaystyle {\mathcal {M}}_{n}=\left[{\begin{array}{cccc}\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {y} _{n})&\mathbf {GG} ^{\intercal }&\mathbf {f} _{t}(t_{n},\mathbf {y} _{n})&\mathbf {f} (t_{n},\mathbf {y} _{n})\\\mathbf {0} &-\mathbf {f} _{\mathbf {x} }^{\intercal }(t_{n},\mathbf {y} _{n})&\mathbf {0} &\mathbf {0} \\\mathbf {0} &\mathbf {0} &0&1\\\mathbf {0} &\mathbf {0} &0&0\end{array}}\right]\in \mathbb {R} ^{(2d+2)\times (2d+2)},}" loading="lazy">

</div>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{\mathbf {\xi } _{n}\}}</annotation>

</semantics>

</math><img src="./3013d67d5b292b64d9cabe1f1b9b734b6793d9fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.562ex; height:2.843ex;" alt="{\displaystyle \{\mathbf {\xi } _{n}\}}" loading="lazy"> is a sequence of d-dimensional independent <a href="Bernoulli_distribution" title="Bernoulli distribution">two-points distributed random vectors</a> satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(\xi _{n}^{k}=\pm 1)={\frac {1}{2}}}">

</mrow>

</mrow>

</msubsup>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(\xi _{n}^{k}=\pm 1)={\frac {1}{2}}}</annotation>

</semantics>

</math><img src="./8f73f9f21c8c9d8c4c108df013f5f70194cc6ca1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.958ex; height:5.176ex;" alt="{\displaystyle P(\xi _{n}^{k}=\pm 1)={\frac {1}{2}}}" loading="lazy">.

<div class="mw-heading mw-heading4"><h4 id="Order_2_WLL_scheme">Order 2 WLL scheme</h4></div>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{n+1}=\mathbf {y} _{n}+\mathbf {B} _{16}+(\mathbf {B} _{14}\mathbf {B} _{11}^{\intercal })^{1/2}\mathbf {\xi } _{n},}</annotation>

</semantics>

</math><img src="./d6c0a22264d44b3ba49b2bdf3daaba84b12ce27f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.862ex; height:3.509ex;" alt="{\displaystyle \mathbf {y} _{n+1}=\mathbf {y} _{n}+\mathbf {B} _{16}+(\mathbf {B} _{14}\mathbf {B} _{11}^{\intercal })^{1/2}\mathbf {\xi } _{n},}" loading="lazy">

</div>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {B} _{11}}</annotation>

</semantics>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {B} _{14}}</annotation>

</semantics>

</math><img src="./7a7a7a9c63a875256955242ec6f03b5018f6e581.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.777ex; height:2.509ex;" alt="{\displaystyle \mathbf {B} _{14}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} _{16}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {B} _{16}}</annotation>

</semantics>

</math><img src="./70f737d850114653cadec84a25c10668d031cf1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.777ex; height:2.509ex;" alt="{\displaystyle \mathbf {B} _{16}}" loading="lazy"> are the submatrices defined by the partitioned matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} =\mathbf {P} _{p,q}(2^{-k_{n}}{\mathcal {M}}_{n}h_{n}))^{2^{k_{n}}}}">

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {B} =\mathbf {P} _{p,q}(2^{-k_{n}}{\mathcal {M}}_{n}h_{n}))^{2^{k_{n}}}}</annotation>

</semantics>

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

<msup>

</mrow>

</mrow>

</msup>

</mtd>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

<mtd>

<msup>

</mrow>

</mrow>

</msup>

</mtd>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

<mtd>

<msup>

</mrow>

</mrow>

</msup>

</mtd>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}_{n}=\left[{\begin{array}{cccccc}\mathbf {J} &\mathbf {H} _{2}&\mathbf {H} _{1}&\mathbf {H} _{0}&\mathbf {a} _{2}&\mathbf {a} _{1}\\\mathbf {0} &-\mathbf {J} ^{\intercal }&\mathbf {I} &\mathbf {0} &\mathbf {0} &\mathbf {0} \\\mathbf {0} &\mathbf {0} &-\mathbf {J} ^{\intercal }&\mathbf {I} &\mathbf {0} &\mathbf {0} \\\mathbf {0} &\mathbf {0} &\mathbf {0} &-\mathbf {J} ^{\intercal }&\mathbf {0} &\mathbf {0} \\\mathbf {0} &\mathbf {0} &\mathbf {0} &\mathbf {0} &0&1\\\mathbf {0} &\mathbf {0} &\mathbf {0} &\mathbf {0} &0&0\end{array}}\right]\in \mathbb {R} ^{(4d+2)\times (4d+2)},}</annotation>

</semantics>

</math><img src="./e92d5869bcb0a0b5db4944c8170713c0c5a096a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:58.466ex; height:19.176ex;" alt="{\displaystyle {\mathcal {M}}_{n}=\left[{\begin{array}{cccccc}\mathbf {J} &\mathbf {H} _{2}&\mathbf {H} _{1}&\mathbf {H} _{0}&\mathbf {a} _{2}&\mathbf {a} _{1}\\\mathbf {0} &-\mathbf {J} ^{\intercal }&\mathbf {I} &\mathbf {0} &\mathbf {0} &\mathbf {0} \\\mathbf {0} &\mathbf {0} &-\mathbf {J} ^{\intercal }&\mathbf {I} &\mathbf {0} &\mathbf {0} \\\mathbf {0} &\mathbf {0} &\mathbf {0} &-\mathbf {J} ^{\intercal }&\mathbf {0} &\mathbf {0} \\\mathbf {0} &\mathbf {0} &\mathbf {0} &\mathbf {0} &0&1\\\mathbf {0} &\mathbf {0} &\mathbf {0} &\mathbf {0} &0&0\end{array}}\right]\in \mathbb {R} ^{(4d+2)\times (4d+2)},}" loading="lazy">

</div>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {J} =\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {y} _{n})\qquad \mathbf {a} _{1}=\mathbf {f} (t_{n},\mathbf {y} _{n})\qquad \mathbf {a} _{2}=\mathbf {f} _{t}(t_{n},\mathbf {y} _{n})+{\frac {1}{2}}\sum \limits _{i=1}^{m}(\mathbf {I} \otimes (\mathbf {g} ^{i}(t_{n}))^{\intercal })\mathbf {f} _{\mathbf {xx} }(t_{n},\mathbf {y} _{n})\mathbf {g} ^{i}(t_{n})}</annotation>

</semantics>

</math><img src="./fa1b215bee23ea42d1eb2e9e9e2dbed7b26023da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:91.791ex; height:6.843ex;" alt="{\displaystyle \mathbf {J} =\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {y} _{n})\qquad \mathbf {a} _{1}=\mathbf {f} (t_{n},\mathbf {y} _{n})\qquad \mathbf {a} _{2}=\mathbf {f} _{t}(t_{n},\mathbf {y} _{n})+{\frac {1}{2}}\sum \limits _{i=1}^{m}(\mathbf {I} \otimes (\mathbf {g} ^{i}(t_{n}))^{\intercal })\mathbf {f} _{\mathbf {xx} }(t_{n},\mathbf {y} _{n})\mathbf {g} ^{i}(t_{n})}" loading="lazy">

</div>

and

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {H} _{0}=\mathbf {G} (t_{n})\mathbf {G} ^{\intercal }(t_{n})\qquad \mathbf {H} _{1}=\mathbf {G} (t_{n}){\frac {d\mathbf {G} ^{\intercal }(t_{n})}{dt}}+{\frac {d\mathbf {G} (t_{n})}{dt}}\mathbf {G} ^{\intercal }(t_{n})\qquad \mathbf {H} _{2}={\frac {d\mathbf {G} (t_{n})}{dt}}{\frac {d\mathbf {G} ^{\intercal }(t_{n})}{dt}}{\text{.}}}</annotation>

</semantics>

</math><img src="./9c72edd9b6adebd3676490c2949bca3ab996a99c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:92.048ex; height:5.843ex;" alt="{\displaystyle \mathbf {H} _{0}=\mathbf {G} (t_{n})\mathbf {G} ^{\intercal }(t_{n})\qquad \mathbf {H} _{1}=\mathbf {G} (t_{n}){\frac {d\mathbf {G} ^{\intercal }(t_{n})}{dt}}+{\frac {d\mathbf {G} (t_{n})}{dt}}\mathbf {G} ^{\intercal }(t_{n})\qquad \mathbf {H} _{2}={\frac {d\mathbf {G} (t_{n})}{dt}}{\frac {d\mathbf {G} ^{\intercal }(t_{n})}{dt}}{\text{.}}}" loading="lazy">

</div>

<div class="mw-heading mw-heading4"><h4 id="Stability_and_dynamics_3">Stability and dynamics</h4></div>

By construction, the weak LL discretizations inherit the stability and <a href="Random_dynamical_system" title="Random dynamical system">dynamics</a> of the linear SDEs, but it is not the case of the weak LL schemes in general. WLL schemes, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\leq q\leq p+2,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\leq q\leq p+2,}</annotation>

</semantics>

</math><img src="./b791ac14adebd1529487660fe99c5a70dcc89f10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:14.344ex; height:2.509ex;" alt="{\displaystyle p\leq q\leq p+2,}" loading="lazy"> preserve the <a href="Moment_(mathematics)" title="Moment (mathematics)">first two moments</a> of the linear SDEs, and inherits the mean-square stability or instability that such solution may have.<a href="#cite_note-:11-24">[24]</a> This includes, for instance, the equations of coupled harmonic oscillators driven by random force, and large systems of stiff linear SDEs that result from the method of lines for linear stochastic partial differential equations. Moreover, these WLL schemes preserve the <a href="Ergodicity" title="Ergodicity">ergodicity</a> of the linear equations, and are geometrically ergodic for some classes of nonlinear SDEs.<a href="#cite_note-:6-26">[26]</a> For nonlinear SDEs with small noise (i.e., (8.1) with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {g} _{i}(t)\approx 0}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {g} _{i}(t)\approx 0}</annotation>

</semantics>

</math><img src="./d413c2850a8e0526f4d32dfa70782308c627f3da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.046ex; height:2.843ex;" alt="{\displaystyle \mathbf {g} _{i}(t)\approx 0}" loading="lazy">), the solutions of these WLL schemes are basically the nonrandom paths of the LL scheme (4.6) for ODEs plus a small disturbance related to the small noise. In this situation, the dynamical properties of that deterministic scheme, such as the linearization preserving and the preservation of the exact solution dynamics around hyperbolic equilibrium points and periodic orbits, become relevant for the mean of the WLL scheme.<a href="#cite_note-:11-24">[24]</a> For instance, Fig. 5 shows the approximate mean of the SDE

<msup>

</mrow>

</msup>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dx=-t^{2}x{\text{ }}dt+{\frac {3}{2(t+1)}}e^{-t^{3}/3}{\text{ }}dw_{t},\qquad \qquad x(0)=1,\qquad \quad (8.2)}</annotation>

</semantics>

</math><img src="./c9e8b65cacf7696f41dfa5f1e5ca7111b8de72ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:66.422ex; height:6.009ex;" alt="{\displaystyle dx=-t^{2}x{\text{ }}dt+{\frac {3}{2(t+1)}}e^{-t^{3}/3}{\text{ }}dw_{t},\qquad \qquad x(0)=1,\qquad \quad (8.2)}" loading="lazy">

computed by various schemes.

<div class="mw-heading mw-heading2"><h2 id="Historical_notes">Historical notes</h2></div>

Below is a time line of the main developments of the Local Linearization (LL) method.

<ul><li>Pope D.A. (1963) introduces the LL discretization for ODEs and the LL scheme based on Taylor expansion.<a href="#cite_note-:18-2">[2]</a></li>

<li>Ozaki T. (1985) introduces the LL method for the integration and estimation of SDEs. The term "Local Linearization" is used for first time.<a href="#cite_note-:19-27">[27]</a></li>

<li>Biscay R. et al. (1996) reformulate the strong LL method for SDEs.<a href="#cite_note-:20-19">[19]</a></li>

<li>Shoji I. and Ozaki T. (1997) reformulate the weak LL method for SDEs.<a href="#cite_note-:21-23">[23]</a></li>

<li>Hochbruck M. et al. (1998) introduce the LL scheme for ODEs based on Krylov subspace approximation.<a href="#cite_note-:22-3">[3]</a></li>

<li>Jimenez J.C. (2002) introduces the LL scheme for ODEs and SDEs based on rational Padé approximation.<a href="#cite_note-:23-21">[21]</a></li>

<li>Carbonell F.M. et al. (2005) introduce the LL method for RDEs.<a href="#cite_note-:24-16">[16]</a></li>

<li>Jimenez J.C. et al. (2006) introduce the LL method for DDEs.<a href="#cite_note-:13-11">[11]</a></li>

<li>De la Cruz H. et al. (2006, 2007) and Tokman M. (2006) introduce the two classes of HOLL integrators for ODEs: the integrator-based <a href="#cite_note-:1-6">[6]</a> and the quadrature-based.<a href="#cite_note-:17-7">[7]</a><a href="#cite_note-:2-5">[5]</a></li>

<li>De la Cruz H. et al. (2010) introduce strong HOLL method for SDEs.<a href="#cite_note-:4-20">[20]</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist">

<li id="cite_note-:8-1">^ <a href="#cite_ref-:8_1-0">a</a> <a href="#cite_ref-:8_1-1">b</a> <a href="#cite_ref-:8_1-2">c</a> <a href="#cite_ref-:8_1-3">d</a> Jimenez J.C. (2009). "Local Linearization methods for the numerical integration of ordinary differential equations: An overview". <a rel="nofollow" class="external text" href="https://inis.iaea.org/search/search.aspx?orig_q=RN:40101978">ICTP Technical Report</a>. 035: 357–373.

</li>

<li id="cite_note-:18-2">^ <a href="#cite_ref-:18_2-0">a</a> <a href="#cite_ref-:18_2-1">b</a> Pope, D. A. (1963). "An exponential method of numerical integration of ordinary differential equations". Comm. ACM, 6(8), 491-493. <a href="https://doi.org/10.1145/366707.367592" class="extiw external" title="doi:10.1145/366707.367592">doi:10.1145/366707.367592</a>.

</li>

<li id="cite_note-:22-3">^ <a href="#cite_ref-:22_3-0">a</a> <a href="#cite_ref-:22_3-1">b</a> <a href="#cite_ref-:22_3-2">c</a> Hochbruck, M., Lubich, C., & Selhofer, H. (1998). "Exponential integrators for large systems of differential equations". SIAM J. Scient. Comput. 19(5), 1552-1574. <a href="https://doi.org/10.1137/S1064827595295337" class="extiw external" title="doi:10.1137/S1064827595295337">doi:10.1137/S1064827595295337</a>.

</li>

<li id="cite_note-:3-4">^ <a href="#cite_ref-:3_4-0">a</a> <a href="#cite_ref-:3_4-1">b</a> <a href="#cite_ref-:3_4-2">c</a> <a href="#cite_ref-:3_4-3">d</a> <a href="#cite_ref-:3_4-4">e</a> <a href="#cite_ref-:3_4-5">f</a> <a href="#cite_ref-:3_4-6">g</a> <a href="#cite_ref-:3_4-7">h</a> de la Cruz H.; Biscay R.J.; Jimenez J.C.; Carbonell F. (2013). "Local Linearization - Runge Kutta Methods: a class of A-stable explicit integrators for dynamical systems". Math. Comput. Modelling. 57 (3–4): 720–740. <a href="https://doi.org/10.1016/j.mcm.2012.08.011" class="extiw external" title="doi:10.1016/j.mcm.2012.08.011">doi:10.1016/j.mcm.2012.08.011</a>.

</li>

<li id="cite_note-:2-5">^ <a href="#cite_ref-:2_5-0">a</a> <a href="#cite_ref-:2_5-1">b</a> <a href="#cite_ref-:2_5-2">c</a> <a href="#cite_ref-:2_5-3">d</a> <a href="#cite_ref-:2_5-4">e</a> <a href="#cite_ref-:2_5-5">f</a> <a href="#cite_ref-:2_5-6">g</a> <a href="#cite_ref-:2_5-7">h</a> de la Cruz H.; Biscay R.J.; Carbonell F.; Ozaki T.; Jimenez J.C. (2007). "A higher order Local Linearization method for solving ordinary differential equations". Appl. Math. Comput. 185: 197–212. <a href="https://doi.org/10.1016/j.amc.2006.06.096" class="extiw external" title="doi:10.1016/j.amc.2006.06.096">doi:10.1016/j.amc.2006.06.096</a>.

</li>

<li id="cite_note-:1-6">^ <a href="#cite_ref-:1_6-0">a</a> <a href="#cite_ref-:1_6-1">b</a> <a href="#cite_ref-:1_6-2">c</a> <a href="#cite_ref-:1_6-3">d</a> <a href="#cite_ref-:1_6-4">e</a> de la Cruz H.; Biscay R.J.; Carbonell F.; Jimenez J.C.; Ozaki T. (2006). "Local Linearization-Runge Kutta (LLRK) methods for solving ordinary differential equations". Lecture Note in Computer Sciences 3991: 132–139, Springer-Verlag. <a href="https://doi.org/10.1007/11758501_22" class="extiw external" title="doi:10.1007/11758501 22">doi:10.1007/11758501 22</a>. <style data-mw-deduplicate="TemplateStyles:r1238218222">

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-34379-0</bdi>.

</li>

<li id="cite_note-:17-7">^ <a href="#cite_ref-:17_7-0">a</a> <a href="#cite_ref-:17_7-1">b</a> Tokman M. (2006). "Efficient integration of large stiff systems of ODEs with exponential propagation iterative (EPI) methods". J. Comput. Physics. 213 (2): 748–776. <a href="https://doi.org/10.1016/j.jcp.2005.08.032" class="extiw external" title="doi:10.1016/j.jcp.2005.08.032">doi:10.1016/j.jcp.2005.08.032</a>.

</li>

<li id="cite_note-:7-8"><a href="#cite_ref-:7_8-0">^</a> M. Hochbruck.; A. Ostermann. (2011). "Exponential multistep methods of Adams-type". BIT Numer. Math. 51 (4): 889–908. <a href="https://doi.org/10.1007/s10543-011-0332-6" class="extiw external" title="doi:10.1007/s10543-011-0332-6">doi:10.1007/s10543-011-0332-6</a>.

</li>

<li id="cite_note-:25-9">^ <a href="#cite_ref-:25_9-0">a</a> <a href="#cite_ref-:25_9-1">b</a> <a href="#cite_ref-:25_9-2">c</a> <a href="#cite_ref-:25_9-3">d</a> <a href="#cite_ref-:25_9-4">e</a> Jimenez, J. C., & Carbonell, F. (2005). "Rate of convergence of local linearization schemes for initial-value problems". Appl. Math. Comput., 171(2), 1282-1295. <a href="https://doi.org/10.1016/j.amc.2005.01.118" class="extiw external" title="doi:10.1016/j.amc.2005.01.118">doi:10.1016/j.amc.2005.01.118</a>.

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> Carbonell F.; Jimenez J.C.; Pedroso L.M. (2008). "Computing multiple integrals involving matrix exponentials". J. Comput. Appl. Math. 213: 300–305. <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.cam.2007.01.007">doi:10.1016/j.cam.2007.01.007</a>.

</li>

<li id="cite_note-:13-11">^ <a href="#cite_ref-:13_11-0">a</a> <a href="#cite_ref-:13_11-1">b</a> <a href="#cite_ref-:13_11-2">c</a> <a href="#cite_ref-:13_11-3">d</a> Jimenez J.C.; Pedroso L.; Carbonell F.; Hernandez V. (2006). "Local linearization method for numerical integration of delay differential equations". SIAM J. Numer. Analysis. 44 (6): 2584–2609. <a href="https://doi.org/10.1137/040607356" class="extiw external" title="doi:10.1137/040607356">doi:10.1137/040607356</a>.

</li>

<li id="cite_note-:12-12">^ <a href="#cite_ref-:12_12-0">a</a> <a href="#cite_ref-:12_12-1">b</a> <a href="#cite_ref-:12_12-2">c</a> <a href="#cite_ref-:12_12-3">d</a> <a href="#cite_ref-:12_12-4">e</a> <a href="#cite_ref-:12_12-5">f</a> Jimenez J.C.; de la Cruz H. (2012). "Convergence rate of strong Local Linearization schemes for stochastic differential equations with additive noise". BIT Numer. Math. 52 (2): 357–382. <a href="https://doi.org/10.1007/s10543-011-0360-2" class="extiw external" title="doi:10.1007/s10543-011-0360-2">doi:10.1007/s10543-011-0360-2</a>.

</li>

<li id="cite_note-:9-13">^ <a href="#cite_ref-:9_13-0">a</a> <a href="#cite_ref-:9_13-1">b</a> <a href="#cite_ref-:9_13-2">c</a> Jimenez J.C.; Biscay R.; Mora C.; Rodriguez L.M. (2002). "Dynamic properties of the Local Linearization method for initial-value problems". Appl. Math. Comput. 126: 63–68. <a href="https://doi.org/10.1016/S0096-3003(00)00100-4" class="extiw external" title="doi:10.1016/S0096-3003(00)00100-4">doi:10.1016/S0096-3003(00)00100-4</a>.

</li>

<li id="cite_note-:15-14"><a href="#cite_ref-:15_14-0">^</a> Jimenez J.C.; Sotolongo A.; Sanchez-Bornot J.M. (2014). "Locally Linearized Runge Kutta method of Dormand and Prince". Appl. Math. Comput. 247: 589–606. <a href="https://doi.org/10.1016/j.amc.2014.09.001" class="extiw external" title="doi:10.1016/j.amc.2014.09.001">doi:10.1016/j.amc.2014.09.001</a>.

</li>

<li id="cite_note-:16-15"><a href="#cite_ref-:16_15-0">^</a> Naranjo-Noda, Jimenez J.C. (2021) "Locally Linearized Runge_Kutta method of Dormand and Prince for large systems of initial value problems." J.Comput. Physics. 426: 109946. <a href="https://doi.org/10.1016/j.jcp.2020.109946" class="extiw external" title="doi:10.1016/j.jcp.2020.109946">doi:10.1016/j.jcp.2020.109946</a>.

</li>

<li id="cite_note-:24-16">^ <a href="#cite_ref-:24_16-0">a</a> <a href="#cite_ref-:24_16-1">b</a> <a href="#cite_ref-:24_16-2">c</a> Carbonell, F., Jimenez, J. C., Biscay, R. J., & De La Cruz, H. (2005). "The local linearization method for numerical integration of random differential equations". BIT Num. Math. 45(1), 1-14. <a href="https://doi.org/10.1007/S10543-005-2645-9" class="extiw external" title="doi:10.1007/S10543-005-2645-9">doi:10.1007/S10543-005-2645-9</a>.

</li>

<li id="cite_note-:10-17">^ <a href="#cite_ref-:10_17-0">a</a> <a href="#cite_ref-:10_17-1">b</a> Jimenez J.C.; Carbonell F. (2009). "Rate of convergence of local linearization schemes for random differential equations". BIT Numer. Math. 49 (2): 357–373. <a href="https://doi.org/10.1007/s10543-009-0225-0" class="extiw external" title="doi:10.1007/s10543-009-0225-0">doi:10.1007/s10543-009-0225-0</a>.

</li>

<li id="cite_note-:14-18"><a href="#cite_ref-:14_18-0">^</a> Jimenez J.C, Shoji I., Ozaki T. (1999) "Simulación of stochastic differential equation through the local linearization method. A comparative study". J. Statist. Physics. 99: 587-602, <a href="https://doi.org/10.1023/A:1004504506041" class="extiw external" title="doi:10.1023/A:1004504506041">doi:10.1023/A:1004504506041</a>.

</li>

<li id="cite_note-:20-19">^ <a href="#cite_ref-:20_19-0">a</a> <a href="#cite_ref-:20_19-1">b</a> Biscay, R., Jimenez, J. C., Riera, J. J., & Valdes, P. A. (1996). "Local linearization method for the numerical solution of stochastic differential equations". Annals Inst. Statis. Math. 48(4), 631-644. <a href="https://doi.org/10.1007/BF00052324" class="extiw external" title="doi:10.1007/BF00052324">doi:10.1007/BF00052324</a>.

</li>

<li id="cite_note-:4-20">^ <a href="#cite_ref-:4_20-0">a</a> <a href="#cite_ref-:4_20-1">b</a> <a href="#cite_ref-:4_20-2">c</a> <a href="#cite_ref-:4_20-3">d</a> <a href="#cite_ref-:4_20-4">e</a> <a href="#cite_ref-:4_20-5">f</a> <a href="#cite_ref-:4_20-6">g</a> de la Cruz H.; Biscay R.J.; Jimenez J.C.; Carbonell F.; Ozaki T. (2010). "High Order Local Linearization methods: an approach for constructing A-stable high order explicit schemes for stochastic differential equations with additive noise". BIT Numer. Math. 50 (3): 509–539. <a href="https://doi.org/10.1007/s10543-010-0272-6" class="extiw external" title="doi:10.1007/s10543-010-0272-6">doi:10.1007/s10543-010-0272-6</a>.

</li>

<li id="cite_note-:23-21">^ <a href="#cite_ref-:23_21-0">a</a> <a href="#cite_ref-:23_21-1">b</a> <a href="#cite_ref-:23_21-2">c</a> Jimenez, J. C. (2002). "A simple algebraic expression to evaluate the local linearization schemes for stochastic differential equations". Appl. Math. Letters, 15(6), 775-780. <a href="https://doi.org/10.1016/S0893-9659(02)00041-1" class="extiw external" title="doi:10.1016/S0893-9659(02)00041-1">doi:10.1016/S0893-9659(02)00041-1</a>.

</li>

<li id="cite_note-:5-22"><a href="#cite_ref-:5_22-0">^</a> de la Cruz H.; Jimenez J.C.; Zubelli J.P. (2017). "Locally Linearized methods for the simulation of stochastic oscillators driven by random forces". BIT Numer. Math. 57: 123–151. <a href="https://doi.org/10.1007/s10543-016-0620-2" class="extiw external" title="doi:10.1007/s10543-016-0620-2">doi:10.1007/s10543-016-0620-2</a>.

</li>

<li id="cite_note-:21-23">^ <a href="#cite_ref-:21_23-0">a</a> <a href="#cite_ref-:21_23-1">b</a> Shoji, I., & Ozaki, T. (1997). "Comparative study of estimation methods for continuous time stochastic processes". J. Time Series Anal. 18(5), 485-506. <a href="https://doi.org/10.1111/1467-9892.00064" class="extiw external" title="doi:10.1111/1467-9892.00064">doi:10.1111/1467-9892.00064</a>.

</li>

<li id="cite_note-:11-24">^ <a href="#cite_ref-:11_24-0">a</a> <a href="#cite_ref-:11_24-1">b</a> <a href="#cite_ref-:11_24-2">c</a> <a href="#cite_ref-:11_24-3">d</a> Jimenez J.C.; Carbonell F. (2015). "Convergence rate of weak Local Linearization schemes for stochastic differential equations with additive noise". J. Comput. Appl. Math. 279: 106–122. <a href="https://doi.org/10.1016/j.cam.2014.10.021" class="extiw external" title="doi:10.1016/j.cam.2014.10.021">doi:10.1016/j.cam.2014.10.021</a>.

</li>

<li id="cite_note-:0-25">^ <a href="#cite_ref-:0_25-0">a</a> <a href="#cite_ref-:0_25-1">b</a> Carbonell F.; Jimenez J.C.; Biscay R.J. (2006). "Weak local linear discretizations for stochastic differential equations: convergence and numerical schemes". J. Comput. Appl. Math. 197: 578–596. <a href="https://doi.org/10.1016/j.cam.2005.11.032" class="extiw external" title="doi:10.1016/j.cam.2005.11.032">doi:10.1016/j.cam.2005.11.032</a>.

</li>

<li id="cite_note-:6-26"><a href="#cite_ref-:6_26-0">^</a> Hansen N.R. (2003) "Geometric ergodicity of discre-time approximations to multivariate diffusion". Bernoulli. 9 : 725-743, <a href="https://doi.org/10.3150/bj/1066223276" class="extiw external" title="doi:10.3150/bj/1066223276">doi:10.3150/bj/1066223276</a>.

</li>

<li id="cite_note-:19-27"><a href="#cite_ref-:19_27-0">^</a> Ozaki, T. (1985). "Non-linear time series models and dynamical systems". Handbook of statistics, 5, 25-83. <a href="https://doi.org/10.1016/S0169-7161(85)05004-0" class="extiw external" title="doi:10.1016/S0169-7161(85)05004-0">doi:10.1016/S0169-7161(85)05004-0</a>.

</li>

</ol></div></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-04-14" href="https://en.wikipedia.org/wiki/?title=Local_linearization_method&oldid=1285633863">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>