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<title>Linear multistep method</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Linear multistep method</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">"Adams' method" redirects here. For the electoral apportionment method, see <a href="Method_of_smallest_divisors" class="mw-redirect" title="Method of smallest divisors">Method of smallest divisors</a>.</div>

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<p><b>Linear multistep methods</b> are used for the <a href="Numerical_ordinary_differential_equations" class="mw-redirect" title="Numerical ordinary differential equations">numerical solution of ordinary differential equations</a>. Conceptually, a numerical method starts from an initial point and then takes a short <b>step</b> forward in time to find the next solution point. The process continues with subsequent steps to map out the solution. Single-step methods (such as <a href="Euler's_method" class="mw-redirect" title="Euler's method">Euler's method</a>) refer to only one previous point and its derivative to determine the current value. Methods such as <a href="Runge%E2%80%93Kutta_methods" title="Runge–Kutta methods">Runge–Kutta</a> take some intermediate steps (for example, a half-step) to obtain a higher order method, but then discard all previous information before taking a second step. Multistep methods attempt to gain efficiency by keeping and using the information from previous steps rather than discarding it. Consequently, multistep methods refer to several previous points and derivative values. In the case of <i>linear</i> multistep methods, a <a href="Linear_combination" title="Linear combination">linear combination</a> of the previous points and derivative values is used.

</p>

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

<p>Numerical methods for ordinary differential equations approximate solutions to <a href="Initial_value_problem" title="Initial value problem">initial value problems</a> of the form

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<annotation encoding="application/x-tex">{\displaystyle y'=f(t,y),\quad y(t_{0})=y_{0}.}</annotation>

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</p><p>The result is approximations for the value of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(t)}">

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

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</math></span><img src="./397de1edef5bf2ee15c020f325d7d781a3aa7f50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.804ex; height:2.843ex;" alt="{\displaystyle y(t)}" loading="lazy"></span> at discrete times <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{i}}">

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

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</math></span><img src="./8b61e3d4d909be4a19c9a554a301684232f59e5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.639ex; height:2.343ex;" alt="{\displaystyle t_{i}}" loading="lazy"></span>:

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<annotation encoding="application/x-tex">{\displaystyle y_{i}\approx y(t_{i})\quad {\text{where}}\quad t_{i}=t_{0}+ih,}</annotation>

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where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

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

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</math></span><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"></span> is the time step (sometimes referred to as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t}">

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

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</math></span><img src="./8c28867ecd34e2caed12cf38feadf6a81a7ee542.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.775ex; height:2.176ex;" alt="{\displaystyle \Delta t}" loading="lazy"></span>) and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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

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</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span> is an integer.

</p><p>Multistep methods use information from the previous <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

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

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</math></span><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"></span> steps to calculate the next value. In particular, a <i>linear</i> multistep method uses a linear combination of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}}">

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

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</math></span><img src="./67d30d30b6c2dbe4d6f150d699de040937ecc95f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.009ex;" alt="{\displaystyle y_{i}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t_{i},y_{i})}">

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<annotation encoding="application/x-tex">{\displaystyle f(t_{i},y_{i})}</annotation>

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</math></span><img src="./b355677d88f29edd15c0620aa7ecc57c6f58e143.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.7ex; height:2.843ex;" alt="{\displaystyle f(t_{i},y_{i})}" loading="lazy"></span> to calculate the value of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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

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</math></span><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span> for the desired current step. Thus, a linear multistep method is a method of the form

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&y_{n+s}+a_{s-1}\cdot y_{n+s-1}+a_{s-2}\cdot y_{n+s-2}+\cdots +a_{0}\cdot y_{n}\\&\qquad {}=h\cdot \left(b_{s}\cdot f(t_{n+s},y_{n+s})+b_{s-1}\cdot f(t_{n+s-1},y_{n+s-1})+\cdots +b_{0}\cdot f(t_{n},y_{n})\right)\\&\Leftrightarrow \sum _{j=0}^{s}a_{j}y_{n+j}=h\sum _{j=0}^{s}b_{j}f(t_{n+j},y_{n+j}),\end{aligned}}}</annotation>

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with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{s}=1}">

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<annotation encoding="application/x-tex">{\displaystyle a_{s}=1}</annotation>

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</math></span><img src="./69cecc0fb9a9891dc82c8b3fc19258894ce6018a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.494ex; height:2.509ex;" alt="{\displaystyle a_{s}=1}" loading="lazy"></span>. The coefficients <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0},\dotsc ,a_{s-1}}">

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<annotation encoding="application/x-tex">{\displaystyle a_{0},\dotsc ,a_{s-1}}</annotation>

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</math></span><img src="./56eb11bd96651801f37f83ffc939d3e4f9d8b04a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.796ex; height:2.009ex;" alt="{\displaystyle a_{0},\dotsc ,a_{s-1}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0},\dotsc ,b_{s}}">

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<annotation encoding="application/x-tex">{\displaystyle b_{0},\dotsc ,b_{s}}</annotation>

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</math></span><img src="./4af2f035e89a559b4caac036d66abb3cc64b81c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.231ex; height:2.509ex;" alt="{\displaystyle b_{0},\dotsc ,b_{s}}" loading="lazy"></span> determine the method. The designer of the method chooses the coefficients, balancing the need to get a good approximation to the true solution against the desire to get a method that is easy to apply. Often, many coefficients are zero to simplify the method.

</p><p>One can distinguish between <a href="Explicit_and_implicit_methods" title="Explicit and implicit methods">explicit and implicit methods</a>. If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{s}=0}">

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<annotation encoding="application/x-tex">{\displaystyle b_{s}=0}</annotation>

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</math></span><img src="./2d4d5de1ceefce6588004f40ab946282a620ed54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.262ex; height:2.509ex;" alt="{\displaystyle b_{s}=0}" loading="lazy"></span>, then the method is called "explicit", since the formula can directly compute <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{n+s}}">

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

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</math></span><img src="./9d2e92e61a539e7de602f3ec6b849f78ba309c50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.407ex; height:2.009ex;" alt="{\displaystyle y_{n+s}}" loading="lazy"></span>. If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{s}\neq 0}">

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<annotation encoding="application/x-tex">{\displaystyle b_{s}\neq 0}</annotation>

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</math></span><img src="./6589d429898a12d1bf32d4a31c87343b75ccd857.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.262ex; height:2.676ex;" alt="{\displaystyle b_{s}\neq 0}" loading="lazy"></span> then the method is called "implicit", since the value of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{n+s}}">

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

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</math></span><img src="./9d2e92e61a539e7de602f3ec6b849f78ba309c50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.407ex; height:2.009ex;" alt="{\displaystyle y_{n+s}}" loading="lazy"></span> depends on the value of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t_{n+s},y_{n+s})}">

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<annotation encoding="application/x-tex">{\displaystyle f(t_{n+s},y_{n+s})}</annotation>

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</math></span><img src="./e9e67b5a5d496c0b0e81747df48941b0396f6635.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.637ex; height:2.843ex;" alt="{\displaystyle f(t_{n+s},y_{n+s})}" loading="lazy"></span>, and the equation must be solved for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{n+s}}">

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

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</math></span><img src="./9d2e92e61a539e7de602f3ec6b849f78ba309c50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.407ex; height:2.009ex;" alt="{\displaystyle y_{n+s}}" loading="lazy"></span>. <a href="Iterative_methods" class="mw-redirect" title="Iterative methods">Iterative methods</a> such as <a href="Newton's_method" title="Newton's method">Newton's method</a> are often used to solve the implicit formula.

</p><p>Sometimes an explicit multistep method is used to "predict" the value of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{n+s}}">

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

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</math></span><img src="./9d2e92e61a539e7de602f3ec6b849f78ba309c50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.407ex; height:2.009ex;" alt="{\displaystyle y_{n+s}}" loading="lazy"></span>. That value is then used in an implicit formula to "correct" the value. The result is a <a href="Predictor%E2%80%93corrector_method" title="Predictor–corrector method">predictor–corrector method</a>.

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

<p>Consider for an example the problem

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<annotation encoding="application/x-tex">{\displaystyle y'=f(t,y)=y,\quad y(0)=1.}</annotation>

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The exact solution is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(t)=e^{t}}">

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

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<div class="mw-heading mw-heading3"><h3 id="One-step_Euler">One-step Euler</h3></div>

<p>A simple numerical method is Euler's method:

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<annotation encoding="application/x-tex">{\displaystyle y_{n+1}=y_{n}+hf(t_{n},y_{n}).}</annotation>

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Euler's method can be viewed as an explicit multistep method for the degenerate case of one step.

</p><p>This method, applied with step size <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h={\tfrac {1}{2}}}">

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<annotation encoding="application/x-tex">{\displaystyle h={\tfrac {1}{2}}}</annotation>

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</math></span><img src="./b0ebad512edec0b270c313d7a7694e57174983b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.096ex; height:3.509ex;" alt="{\displaystyle h={\tfrac {1}{2}}}" loading="lazy"></span> on the problem <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y'=y}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}y_{1}&=y_{0}+hf(t_{0},y_{0})=1+{\tfrac {1}{2}}\cdot 1=1.5,\\y_{2}&=y_{1}+hf(t_{1},y_{1})=1.5+{\tfrac {1}{2}}\cdot 1.5=2.25,\\y_{3}&=y_{2}+hf(t_{2},y_{2})=2.25+{\tfrac {1}{2}}\cdot 2.25=3.375,\\y_{4}&=y_{3}+hf(t_{3},y_{3})=3.375+{\tfrac {1}{2}}\cdot 3.375=5.0625.\end{aligned}}}</annotation>

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</p>

<div class="mw-heading mw-heading3"><h3 id="Two-step_Adams–Bashforth">Two-step Adams–Bashforth</h3></div>

<p>Euler's method is a <a href="One-step_method" title="One-step method">one-step method</a>. A simple multistep method is the two-step Adams–Bashforth method

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<annotation encoding="application/x-tex">{\displaystyle y_{n+2}=y_{n+1}+{\tfrac {3}{2}}hf(t_{n+1},y_{n+1})-{\tfrac {1}{2}}hf(t_{n},y_{n}).}</annotation>

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This method needs two values, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{n+1}}">

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

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</math></span><img src="./6477fbeae2962cc55973c2298b8653cfd4f5e5d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.458ex; height:2.009ex;" alt="{\displaystyle y_{n+1}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{n}}">

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

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</math></span><img src="./2c5fbb0c89590b028eba7239a8803fd0cd2e698e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.358ex; height:2.009ex;" alt="{\displaystyle y_{n}}" loading="lazy"></span>, to compute the next value, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{n+2}}">

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</math></span><img src="./ed0606a3455add6c6896b0c67131147f99aa8e48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.458ex; height:2.009ex;" alt="{\displaystyle y_{n+2}}" loading="lazy"></span>. However, the initial value problem provides only one value, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{0}=1}">

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<annotation encoding="application/x-tex">{\displaystyle y_{0}=1}</annotation>

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</math></span><img src="./3aa87b0b479f0b8902c2f3b25a3069d31f35b00c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.454ex; height:2.509ex;" alt="{\displaystyle y_{0}=1}" loading="lazy"></span>. One possibility to resolve this issue is to use the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{1}}">

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</math></span><img src="./eef4db76d658a98219aca14df06d9869d2b43c42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.009ex;" alt="{\displaystyle y_{1}}" loading="lazy"></span> computed by Euler's method as the second value. With this choice, the Adams–Bashforth method yields (rounded to four digits):

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}y_{2}&=y_{1}+{\tfrac {3}{2}}hf(t_{1},y_{1})-{\tfrac {1}{2}}hf(t_{0},y_{0})=1.5+{\tfrac {3}{2}}\cdot {\tfrac {1}{2}}\cdot 1.5-{\tfrac {1}{2}}\cdot {\tfrac {1}{2}}\cdot 1=2.375,\\y_{3}&=y_{2}+{\tfrac {3}{2}}hf(t_{2},y_{2})-{\tfrac {1}{2}}hf(t_{1},y_{1})=2.375+{\tfrac {3}{2}}\cdot {\tfrac {1}{2}}\cdot 2.375-{\tfrac {1}{2}}\cdot {\tfrac {1}{2}}\cdot 1.5=3.7812,\\y_{4}&=y_{3}+{\tfrac {3}{2}}hf(t_{3},y_{3})-{\tfrac {1}{2}}hf(t_{2},y_{2})=3.7812+{\tfrac {3}{2}}\cdot {\tfrac {1}{2}}\cdot 3.7812-{\tfrac {1}{2}}\cdot {\tfrac {1}{2}}\cdot 2.375=6.0234.\end{aligned}}}</annotation>

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The exact solution at <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=t_{4}=2}">

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</math></span><img src="./a8e8d710c4ca28b889e645052a35e29b81bb518e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.093ex; height:2.509ex;" alt="{\displaystyle t=t_{4}=2}" loading="lazy"></span> is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{2}=7.3891\ldots }">

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<annotation encoding="application/x-tex">{\displaystyle e^{2}=7.3891\ldots }</annotation>

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</math></span><img src="./c418820a947f59a49f3d633b753bf809379980db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.806ex; height:2.676ex;" alt="{\displaystyle e^{2}=7.3891\ldots }" loading="lazy"></span>, so the two-step Adams–Bashforth method is more accurate than Euler's method. This is always the case if the step size is small enough.

</p>

<div class="mw-heading mw-heading2"><h2 id="Families_of_multistep_methods">Families of multistep methods</h2></div>

<p>Three families of linear multistep methods are commonly used: Adams–Bashforth methods, Adams–Moulton methods, and the <a href="Backward_differentiation_formula" title="Backward differentiation formula">backward differentiation formulas</a> (BDFs).

</p>

<div class="mw-heading mw-heading3"><h3 id="Adams–Bashforth_methods">Adams–Bashforth methods</h3></div>

<p>The Adams–Bashforth methods are explicit methods. The coefficients are <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{s-1}=-1}">

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<annotation encoding="application/x-tex">{\displaystyle a_{s-1}=-1}</annotation>

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</math></span><img src="./cc22c088a286bffe2090097dbaf3a9951d855273.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.403ex; height:2.509ex;" alt="{\displaystyle a_{s-1}=-1}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{s-2}=\cdots =a_{0}=0}">

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<annotation encoding="application/x-tex">{\displaystyle a_{s-2}=\cdots =a_{0}=0}</annotation>

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</math></span><img src="./953a2e27823aaad55d4b892de996db942ba0eecc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.799ex; height:2.509ex;" alt="{\displaystyle a_{s-2}=\cdots =a_{0}=0}" loading="lazy"></span>, while the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{j}}">

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</p><p>The Adams–Bashforth methods with <i>s</i> = 1, 2, 3, 4, 5 are (<a href="#CITEREFHairerNørsettWanner1993">Hairer, Nørsett & Wanner 1993</a>, §III.1; <a href="#CITEREFButcher2003">Butcher 2003</a>, p. 103):

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<mtext>(This is the Euler method)</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}y_{n+1}&=y_{n}+hf(t_{n},y_{n}),\qquad {\text{(This is the Euler method)}}\\y_{n+2}&=y_{n+1}+h\left({\frac {3}{2}}f(t_{n+1},y_{n+1})-{\frac {1}{2}}f(t_{n},y_{n})\right),\\y_{n+3}&=y_{n+2}+h\left({\frac {23}{12}}f(t_{n+2},y_{n+2})-{\frac {16}{12}}f(t_{n+1},y_{n+1})+{\frac {5}{12}}f(t_{n},y_{n})\right),\\y_{n+4}&=y_{n+3}+h\left({\frac {55}{24}}f(t_{n+3},y_{n+3})-{\frac {59}{24}}f(t_{n+2},y_{n+2})+{\frac {37}{24}}f(t_{n+1},y_{n+1})-{\frac {9}{24}}f(t_{n},y_{n})\right),\\y_{n+5}&=y_{n+4}+h\left({\frac {1901}{720}}f(t_{n+4},y_{n+4})-{\frac {2774}{720}}f(t_{n+3},y_{n+3})+{\frac {2616}{720}}f(t_{n+2},y_{n+2})-{\frac {1274}{720}}f(t_{n+1},y_{n+1})+{\frac {251}{720}}f(t_{n},y_{n})\right).\end{aligned}}}</annotation>

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</p><p>The coefficients <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{j}}">

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</math></span><img src="./fa56eff4488494085785b7b0d6e2069bd45a3ce5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.907ex; height:2.843ex;" alt="{\displaystyle b_{j}}" loading="lazy"></span> can be determined as follows. Use <a href="Polynomial_interpolation" title="Polynomial interpolation">polynomial interpolation</a> to find the polynomial <i>p</i> of degree <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s-1}">

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<annotation encoding="application/x-tex">{\displaystyle p(t_{n+i})=f(t_{n+i},y_{n+i}),\qquad {\text{for }}i=0,\ldots ,s-1.}</annotation>

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The <a href="Lagrange_polynomial" title="Lagrange polynomial">Lagrange formula</a> for polynomial interpolation yields

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<annotation encoding="application/x-tex">{\displaystyle p(t)=\sum _{j=0}^{s-1}{\frac {(-1)^{s-j-1}f(t_{n+j},y_{n+j})}{j!(s-j-1)!h^{s-1}}}\prod _{i=0 \atop i\neq j}^{s-1}(t-t_{n+i}).}</annotation>

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The polynomial <i>p</i> is locally a good approximation of the right-hand side of the differential equation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y'=f(t,y)}">

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<annotation encoding="application/x-tex">{\displaystyle y'=p(t)}</annotation>

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</math></span><img src="./9e028a4d952534acd276d46bb7f8e5f24907e149.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.762ex; height:3.009ex;" alt="{\displaystyle y'=p(t)}" loading="lazy"></span> instead. This equation can be solved exactly; the solution is simply the integral of <i>p</i>. This suggests taking

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<annotation encoding="application/x-tex">{\displaystyle y_{n+s}=y_{n+s-1}+\int _{t_{n+s-1}}^{t_{n+s}}p(t)\,\mathrm {d} t.}</annotation>

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The Adams–Bashforth method arises when the formula for <i>p</i> is substituted. The coefficients <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle b_{s-j-1}={\frac {(-1)^{j}}{j!(s-j-1)!}}\int _{0}^{1}\prod _{i=0 \atop i\neq j}^{s-1}(u+i)\,\mathrm {d} u,\qquad {\text{for }}j=0,\ldots ,s-1.}</annotation>

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Replacing <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t,y)}">

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</math></span><img src="./6f90c2874286152a886311c702ca1740f7d44b93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.117ex; height:2.843ex;" alt="{\displaystyle f(t,y)}" loading="lazy"></span> by its interpolant <i>p</i> incurs an error of order <i>h</i><sup><i>s</i></sup>, and it follows that the <i>s</i>-step Adams–Bashforth method has indeed order <i>s</i> (<a href="#CITEREFIserles1996">Iserles 1996</a>, §2.1)

</p><p>The Adams–Bashforth methods were designed by <a href="John_Couch_Adams" title="John Couch Adams">John Couch Adams</a> to solve a differential equation modelling <a href="Capillary_action" title="Capillary action">capillary action</a> due to <a href="Francis_Bashforth" title="Francis Bashforth">Francis Bashforth</a>. <a href="#CITEREFBashforth1883">Bashforth (1883)</a> published his theory and Adams' numerical method (<a href="#CITEREFGoldstine1977">Goldstine 1977</a>).

</p>

<div class="mw-heading mw-heading3"><h3 id="Adams–Moulton_methods">Adams–Moulton methods</h3></div>

<p>The Adams–Moulton methods are similar to the Adams–Bashforth methods in that they also have <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{s-1}=-1}">

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<annotation encoding="application/x-tex">{\displaystyle a_{s-2}=\cdots =a_{0}=0}</annotation>

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</math></span><img src="./953a2e27823aaad55d4b892de996db942ba0eecc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.799ex; height:2.509ex;" alt="{\displaystyle a_{s-2}=\cdots =a_{0}=0}" loading="lazy"></span>. Again the <i>b</i> coefficients are chosen to obtain the highest order possible. However, the Adams–Moulton methods are implicit methods. By removing the restriction that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{s}=0}">

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</math></span><img src="./2d4d5de1ceefce6588004f40ab946282a620ed54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.262ex; height:2.509ex;" alt="{\displaystyle b_{s}=0}" loading="lazy"></span>, an <i>s</i>-step Adams–Moulton method can reach order <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s+1}">

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</math></span><img src="./cc3e2e7deb52c40fc144183ba49b8494015640ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.093ex; height:2.343ex;" alt="{\displaystyle s+1}" loading="lazy"></span>, while an <i>s</i>-step Adams–Bashforth methods has only order <i>s</i>.

</p><p>The Adams–Moulton methods with <i>s</i> = 0, 1, 2, 3, 4 are (<a href="#CITEREFHairerNørsettWanner1993">Hairer, Nørsett & Wanner 1993</a>, §III.1; <a href="#CITEREFQuarteroniSaccoSaleri2000">Quarteroni, Sacco & Saleri 2000</a>) listed, where the first two methods are the <a href="Backward_Euler_method" title="Backward Euler method">backward Euler method</a> and the <a href="Trapezoidal_rule_(differential_equations)" title="Trapezoidal rule (differential equations)">trapezoidal rule</a> respectively:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}y_{n}&=y_{n-1}+hf(t_{n},y_{n}),\\y_{n+1}&=y_{n}+{\frac {1}{2}}h\left(f(t_{n+1},y_{n+1})+f(t_{n},y_{n})\right),\\y_{n+2}&=y_{n+1}+h\left({\frac {5}{12}}f(t_{n+2},y_{n+2})+{\frac {8}{12}}f(t_{n+1},y_{n+1})-{\frac {1}{12}}f(t_{n},y_{n})\right),\\y_{n+3}&=y_{n+2}+h\left({\frac {9}{24}}f(t_{n+3},y_{n+3})+{\frac {19}{24}}f(t_{n+2},y_{n+2})-{\frac {5}{24}}f(t_{n+1},y_{n+1})+{\frac {1}{24}}f(t_{n},y_{n})\right),\\y_{n+4}&=y_{n+3}+h\left({\frac {251}{720}}f(t_{n+4},y_{n+4})+{\frac {646}{720}}f(t_{n+3},y_{n+3})-{\frac {264}{720}}f(t_{n+2},y_{n+2})+{\frac {106}{720}}f(t_{n+1},y_{n+1})-{\frac {19}{720}}f(t_{n},y_{n})\right).\end{aligned}}}</annotation>

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</p><p>The derivation of the Adams–Moulton methods is similar to that of the Adams–Bashforth method; however, the interpolating polynomial uses not only the points <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{n-1},\dots ,t_{n-s}}">

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

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</math></span><img src="./5de6e3cbab7bfac7f9d507abdc7d8c3c6ec2e32d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.445ex; height:2.343ex;" alt="{\displaystyle t_{n-1},\dots ,t_{n-s}}" loading="lazy"></span>, as above, but also <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{n}}">

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</math></span><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"></span>. The coefficients are given by

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<annotation encoding="application/x-tex">{\displaystyle b_{s-j}={\frac {(-1)^{j}}{j!(s-j)!}}\int _{0}^{1}\prod _{i=0 \atop i\neq j}^{s}(u+i-1)\,\mathrm {d} u,\qquad {\text{for }}j=0,\ldots ,s.}</annotation>

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</p><p>The Adams–Moulton methods are solely due to <a href="John_Couch_Adams" title="John Couch Adams">John Couch Adams</a>, like the Adams–Bashforth methods. The name of <a href="Forest_Ray_Moulton" title="Forest Ray Moulton">Forest Ray Moulton</a> became associated with these methods because he realized that they could be used in tandem with the Adams–Bashforth methods as a <a href="Predictor-corrector_method" class="mw-redirect" title="Predictor-corrector method">predictor-corrector</a> pair (<a href="#CITEREFMoulton1926">Moulton 1926</a>); <a href="#CITEREFMilne1926">Milne (1926)</a> had the same idea. Adams used <a href="Newton's_method" title="Newton's method">Newton's method</a> to solve the implicit equation (<a href="#CITEREFHairerNørsettWanner1993">Hairer, Nørsett & Wanner 1993</a>, §III.1).

</p>

<div class="mw-heading mw-heading3"><h3 id="Backward_differentiation_formulas_(BDF)">Backward differentiation formulas (BDF)</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Backward_differentiation_formula" title="Backward differentiation formula">Backward differentiation formula</a></div>

<p>The BDF methods are implicit methods with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{s-1}=\cdots =b_{0}=0}">

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<annotation encoding="application/x-tex">{\displaystyle b_{s-1}=\cdots =b_{0}=0}</annotation>

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</math></span><img src="./623e98347fae488dd8cd468f56c825e50de7c3f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.334ex; height:2.509ex;" alt="{\displaystyle b_{s-1}=\cdots =b_{0}=0}" loading="lazy"></span> and the other coefficients chosen such that the method attains order <i>s</i> (the maximum possible). These methods are especially used for the solution of <a href="Stiff_equation" title="Stiff equation">stiff differential equations</a>.

</p>

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

<p>The central concepts in the analysis of linear multistep methods, and indeed any numerical method for differential equations, are <a href="Numerical_ordinary_differential_equations" class="mw-redirect" title="Numerical ordinary differential equations">convergence, order, and stability</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Consistency_and_order">Consistency and order</h3></div>

<p>The first question is whether the method is consistent: is the difference equation

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&a_{s}y_{n+s}+a_{s-1}y_{n+s-1}+a_{s-2}y_{n+s-2}+\cdots +a_{0}y_{n}\\&\qquad {}=h{\bigl (}b_{s}f(t_{n+s},y_{n+s})+b_{s-1}f(t_{n+s-1},y_{n+s-1})+\cdots +b_{0}f(t_{n},y_{n}){\bigr )},\end{aligned}}}</annotation>

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a good approximation of the differential equation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y'=f(t,y)}">

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</math></span><img src="./ac506badf903f0f8a77ad942daa4c00460eb059d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.061ex; height:3.009ex;" alt="{\displaystyle y'=f(t,y)}" loading="lazy"></span>? More precisely, a multistep method is <i>consistent</i> if the <a href="Local_truncation_error" class="mw-redirect" title="Local truncation error">local truncation error</a> goes to zero faster than the step size <i>h</i> as <i>h</i> goes to zero, where the <i>local truncation error</i> is defined to be the difference between the result <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{n+s}}">

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</math></span><img src="./9d2e92e61a539e7de602f3ec6b849f78ba309c50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.407ex; height:2.009ex;" alt="{\displaystyle y_{n+s}}" loading="lazy"></span> of the method, assuming that all the previous values <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{n+s-1},\ldots ,y_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{s-1}a_{k}=-1\quad {\text{and}}\quad \sum _{k=0}^{s}b_{k}=s+\sum _{k=0}^{s-1}ka_{k}.}</annotation>

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</math></span></span>

All the methods mentioned above are consistent (<a href="#CITEREFHairerNørsettWanner1993">Hairer, Nørsett & Wanner 1993</a>, §III.2).

</p><p>If the method is consistent, then the next question is how well the difference equation defining the numerical method approximates the differential equation. A multistep method is said to have <i>order</i> <i>p</i> if the local error is of order <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(h^{p+1})}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(h^{p+1})}</annotation>

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</math></span><img src="./37c924b6c799d01c981735855e99a17ad46c885c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.081ex; height:3.176ex;" alt="{\displaystyle O(h^{p+1})}" loading="lazy"></span> as <i>h</i> goes to zero. This is equivalent to the following condition on the coefficients of the methods:

</mrow>

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</munderover>

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<annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{s-1}a_{k}=-1\quad {\text{and}}\quad q\sum _{k=0}^{s}k^{q-1}b_{k}=s^{q}+\sum _{k=0}^{s-1}k^{q}a_{k}{\text{ for }}q=1,\ldots ,p.}</annotation>

</semantics>

</math></span></span>

The <i>s</i>-step Adams–Bashforth method has order <i>s</i>, while the <i>s</i>-step Adams–Moulton method has order <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s+1}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./cc3e2e7deb52c40fc144183ba49b8494015640ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.093ex; height:2.343ex;" alt="{\displaystyle s+1}" loading="lazy"></span> (<a href="#CITEREFHairerNørsettWanner1993">Hairer, Nørsett & Wanner 1993</a>, §III.2).

</p><p>These conditions are often formulated using the <i>characteristic polynomials</i>

<msup>

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</munderover>

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<annotation encoding="application/x-tex">{\displaystyle \rho (z)=z^{s}+\sum _{k=0}^{s-1}a_{k}z^{k}\quad {\text{and}}\quad \sigma (z)=\sum _{k=0}^{s}b_{k}z^{k}.}</annotation>

</semantics>

</math></span></span>

In terms of these polynomials, the above condition for the method to have order <i>p</i> becomes

<msup>

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</msup>

<msup>

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</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \rho (e^{h})-h\sigma (e^{h})=O(h^{p+1})\quad {\text{as }}h\to 0.}</annotation>

</semantics>

</math></span></span>

In particular, the method is consistent if it has order at least one, which is the case if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (1)=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho (1)=0}</annotation>

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</math></span><img src="./ffeb3078854574cb9d0db2d59720e5eadf869dce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.435ex; height:2.843ex;" alt="{\displaystyle \rho (1)=0}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho '(1)=\sigma (1)}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho '(1)=\sigma (1)}</annotation>

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</math></span><img src="./2a09ac39b1a7176314962d81bb2209aca439cfbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.258ex; height:3.009ex;" alt="{\displaystyle \rho '(1)=\sigma (1)}" loading="lazy"></span>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Stability_and_convergence">Stability and convergence</h3></div>

<p>The numerical solution of a one-step method depends on the initial condition <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{0}}">

<msub>

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</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./6d943dbbb0b56ca750c4d62c5b54b4ae29a773da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.009ex;" alt="{\displaystyle y_{0}}" loading="lazy"></span>, but the numerical solution of an <i>s</i>-step method depend on all the <i>s</i> starting values, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{0},y_{1},\ldots ,y_{s-1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{0},y_{1},\ldots ,y_{s-1}}</annotation>

</semantics>

</math></span><img src="./f6b31948c88d07501260fe428fa8219c94fce0b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.842ex; height:2.009ex;" alt="{\displaystyle y_{0},y_{1},\ldots ,y_{s-1}}" loading="lazy"></span>. It is thus of interest whether the numerical solution is stable with respect to perturbations in the starting values. A linear multistep method is <a href="Zero_stability" title="Zero stability"><i>zero-stable</i></a> for a certain differential equation on a given time interval, if a perturbation in the starting values of size ε causes the numerical solution over that time interval to change by no more than <i>K</i>ε for some value of <i>K</i> which does not depend on the step size <i>h</i>. This is called "zero-stability" because it is enough to check the condition for the differential equation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y'=0}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y'=0}</annotation>

</semantics>

</math></span><img src="./fea04af5db39eeaa411bf8a2c15404482cb547b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.106ex; height:2.843ex;" alt="{\displaystyle y'=0}" loading="lazy"></span> (<a href="#CITEREFSüliMayers2003">Süli & Mayers 2003</a>, p. 332).

</p><p>If the roots of the characteristic polynomial ρ all have modulus less than or equal to 1 and the roots of modulus 1 are of multiplicity 1, we say that the <i>root condition</i> is satisfied. A linear multistep method is zero-stable if and only if the root condition is satisfied (<a href="#CITEREFSüliMayers2003">Süli & Mayers 2003</a>, p. 335).

</p><p>Now suppose that a consistent linear multistep method is applied to a sufficiently smooth differential equation and that the starting values <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{1},\ldots ,y_{s-1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{1},\ldots ,y_{s-1}}</annotation>

</semantics>

</math></span><img src="./17c9b428fb4907dd4f5e21048e15860926414cae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.615ex; height:2.009ex;" alt="{\displaystyle y_{1},\ldots ,y_{s-1}}" loading="lazy"></span> all converge to the initial value <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./6d943dbbb0b56ca750c4d62c5b54b4ae29a773da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.009ex;" alt="{\displaystyle y_{0}}" loading="lazy"></span> as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\to 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h\to 0}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h\to 0}</annotation>

</semantics>

</math></span><img src="./3f15c1e6800f6e3fca9fec99720fc67a13780215.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.116ex; height:2.176ex;" alt="{\displaystyle h\to 0}" loading="lazy"></span> if and only if the method is zero-stable. This result is known as the <i>Dahlquist equivalence theorem</i>, named after <a href="Germund_Dahlquist" title="Germund Dahlquist">Germund Dahlquist</a>; this theorem is similar in spirit to the <a href="Lax_equivalence_theorem" title="Lax equivalence theorem">Lax equivalence theorem</a> for <a href="Finite_difference_method" title="Finite difference method">finite difference methods</a>. Furthermore, if the method has order <i>p</i>, then the <a href="Global_truncation_error" class="mw-redirect" title="Global truncation error">global error</a> (the difference between the numerical solution and the exact solution at a fixed time) is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(h^{p})}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(h^{p})}</annotation>

</semantics>

</math></span><img src="./6c9c07670a9a1a56a7b9a4723e9c3a3081a0a827.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.981ex; height:2.843ex;" alt="{\displaystyle O(h^{p})}" loading="lazy"></span> (<a href="#CITEREFSüliMayers2003">Süli & Mayers 2003</a>, p. 340).

</p><p>Furthermore, if the method is convergent, the method is said to be <i>strongly stable</i> if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=1}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./078535cde78d90bfa1d9fbb2446204593a921d57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.349ex; height:2.176ex;" alt="{\displaystyle z=1}" loading="lazy"></span> is the only root of modulus 1. If it is convergent and all roots of modulus 1 are not repeated, but there is more than one such root, it is said to be <i>relatively stable</i>. Note that 1 must be a root for the method to be convergent; thus convergent methods are always one of these two.

</p><p>To assess the performance of linear multistep methods on <a href="Stiff_equation" title="Stiff equation">stiff equations</a>, consider the linear test equation <i>y'</i> = λ<i>y</i>. A multistep method applied to this differential equation with step size <i>h</i> yields a linear <a href="Recurrence_relation" title="Recurrence relation">recurrence relation</a> with characteristic polynomial

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \pi (z;h\lambda )=(1-h\lambda \beta _{s})z^{s}+\sum _{k=0}^{s-1}(\alpha _{k}-h\lambda \beta _{k})z^{k}=\rho (z)-h\lambda \sigma (z).}</annotation>

</semantics>

</math></span></span>

This polynomial is called the <i>stability polynomial</i> of the multistep method. If all of its roots have modulus less than one then the numerical solution of the multistep method will converge to zero and the multistep method is said to be <i>absolutely stable</i> for that value of <i>h</i>λ. The method is said to be <i>A-stable</i> if it is absolutely stable for all <i>h</i>λ with negative real part. The <i>region of absolute stability</i> is the set of all <i>h</i>λ for which the multistep method is absolutely stable (<a href="#CITEREFSüliMayers2003">Süli & Mayers 2003</a>, pp. 347 & 348). For more details, see the section on <a href="Stiff_equation#Multistep_methods" title="Stiff equation">stiff equations and multistep methods</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Example">Example</h3></div>

<p>Consider the Adams–Bashforth three-step method

<msub>

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</msub>

<msub>

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<mfrac>

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<mfrac>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle y_{n+3}=y_{n+2}+h\left({23 \over 12}f(t_{n+2},y_{n+2})-{4 \over 3}f(t_{n+1},y_{n+1})+{5 \over 12}f(t_{n},y_{n})\right).}</annotation>

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One characteristic polynomial is thus

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \rho (z)=z^{3}-z^{2}=z^{2}(z-1)}</annotation>

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which has roots <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=0,1}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ee8a0365abc9c277a7c062bbfa7c3141efd1a151.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.545ex; height:2.509ex;" alt="{\displaystyle z=0,1}" loading="lazy"></span>, and the conditions above are satisfied. As <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=1}">

</mstyle>

</mrow>

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

</semantics>

</p><p>The other characteristic polynomial is

<mfrac>

</mfrac>

</mrow>

<msup>

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</msup>

<mfrac>

</mfrac>

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<mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (z)={\frac {23}{12}}z^{2}-{\frac {4}{3}}z+{\frac {5}{12}}}</annotation>

</semantics>

</math></span></span>

</p>

<div class="mw-heading mw-heading2"><h2 id="First_and_second_Dahlquist_barriers">First and second Dahlquist barriers</h2></div>

<p>These two results were proved by <a href="Germund_Dahlquist" title="Germund Dahlquist">Germund Dahlquist</a> and represent an important bound for the order of convergence and for the <a href="Stiff_equation#A-stability" title="Stiff equation">A-stability</a> of a linear multistep method. The first Dahlquist barrier was proved in <a href="#CITEREFDahlquist1956">Dahlquist (1956)</a> and the second in <a href="#CITEREFDahlquist1963">Dahlquist (1963)</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="First_Dahlquist_barrier">First Dahlquist barrier</h3></div>

<p>The first Dahlquist barrier states that a zero-stable and linear <i>q</i>-step multistep method cannot attain an order of convergence greater than <i>q</i> + 1 if <i>q</i> is odd and greater than <i>q</i> + 2 if <i>q</i> is even. If the method is also explicit, then it cannot attain an order greater than <i>q</i> (<a href="#CITEREFHairerNørsettWanner1993">Hairer, Nørsett & Wanner 1993</a>, Thm III.3.5).

</p>

<div class="mw-heading mw-heading3"><h3 id="Second_Dahlquist_barrier">Second Dahlquist barrier</h3></div>

<p>The second Dahlquist barrier states that no explicit linear multistep methods are <a href="Stiff_equation#A-stability" title="Stiff equation">A-stable</a>. Further, the maximal order of an (implicit) A-stable linear multistep method is 2. Among the A-stable linear multistep methods of order 2, the trapezoidal rule has the smallest error constant (<a href="#CITEREFDahlquist1963">Dahlquist 1963</a>, Thm 2.1 and 2.2).

</p>

<ul><li><a href="Digital_energy_gain" class="mw-redirect" title="Digital energy gain">Digital energy gain</a></li></ul>

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

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</style><cite id="CITEREFBashforth1883" class="citation cs2">Bashforth, Francis (1883), <i>An Attempt to test the Theories of Capillary Action by comparing the theoretical and measured forms of drops of fluid. With an explanation of the method of integration employed in constructing the tables which give the theoretical forms of such drops, by J. C. Adams</i>, Cambridge</cite><span class="cs1-maint citation-comment"><code class="cs1-code">{{citation}}</code>: CS1 maint: location missing publisher (link)</span>.</li>

<li><cite id="CITEREFButcher2003" class="citation cs2"><a href="John_C._Butcher" title="John C. Butcher">Butcher, John C.</a> (2003), <i>Numerical Methods for Ordinary Differential Equations</i>, John Wiley, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-96758-3</bdi></cite>.</li>

<li><cite id="CITEREFDahlquist1956" class="citation cs2"><a href="Germund_Dahlquist" title="Germund Dahlquist">Dahlquist, Germund</a> (1956), "Convergence and stability in the numerical integration of ordinary differential equations", <i>Mathematica Scandinavica</i>, <b>4</b>: <span class="nowrap">33–</span>53, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.7146%2Fmath.scand.a-10454">10.7146/math.scand.a-10454</a></span></cite>.</li>

<li><cite id="CITEREFDahlquist1963" class="citation cs2"><a href="Germund_Dahlquist" title="Germund Dahlquist">Dahlquist, Germund</a> (1963), "A special stability problem for linear multistep methods", <i>BIT</i>, <b>3</b>: <span class="nowrap">27–</span>43, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01963532">10.1007/BF01963532</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0006-3835">0006-3835</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120241743">120241743</a></cite>.</li>

<li><cite id="CITEREFGoldstine1977" class="citation cs2"><a href="Herman_Goldstine" title="Herman Goldstine">Goldstine, Herman H.</a> (1977), <i>A History of Numerical Analysis from the 16th through the 19th Century</i>, New York: Springer-Verlag, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-90277-7</bdi></cite>.</li>

<li><cite id="CITEREFHairerNørsettWanner1993" class="citation cs2">Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), <i>Solving ordinary differential equations I: Nonstiff problems</i> (2nd ed.), Berlin: Springer Verlag, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-56670-0</bdi></cite>.</li>

<li><cite id="CITEREFHairerWanner1996" class="citation cs2">Hairer, Ernst; Wanner, Gerhard (1996), <i>Solving ordinary differential equations II: Stiff and differential-algebraic problems</i> (2nd ed.), Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-60452-5</bdi></cite>.</li>

<li><cite id="CITEREFIserles1996" class="citation cs2">Iserles, Arieh (1996), <i>A First Course in the Numerical Analysis of Differential Equations</i>, Cambridge University Press, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1996fcna.book.....I">1996fcna.book.....I</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-55655-2</bdi></cite>.</li>

<li><cite id="CITEREFMilne1926" class="citation cs2">Milne, W. E. (1926), "Numerical integration of ordinary differential equations", <i>American Mathematical Monthly</i>, <b>33</b> (9), Mathematical Association of America: <span class="nowrap">455–</span>460, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2299609">10.2307/2299609</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2299609">2299609</a></cite>.</li>

<li><cite id="CITEREFMoulton1926" class="citation cs2"><a href="Forest_Ray_Moulton" title="Forest Ray Moulton">Moulton, Forest R.</a> (1926), <i>New methods in exterior ballistics</i>, University of Chicago Press</cite>.</li>

<li><cite id="CITEREFQuarteroniSaccoSaleri2000" class="citation cs2"><a href="Alfio_Quarteroni" title="Alfio Quarteroni">Quarteroni, Alfio</a>; Sacco, Riccardo; Saleri, Fausto (2000), <i>Matematica Numerica</i>, Springer Verlag, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-88-470-0077-3</bdi></cite>.</li>

<li><cite id="CITEREFSüliMayers2003" class="citation cs2">Süli, Endre; Mayers, David (2003), <i>An Introduction to Numerical Analysis</i>, <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-00794-1</bdi></cite>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><span class="citation mathworld" id="Reference-Mathworld-Adams_Method"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/AdamsMethod.html">"Adams Method"</a>. <i><a href="MathWorld" title="MathWorld">MathWorld</a></i>.</cite></span></li></ul>

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</style><div id="Numerical_methods_for_ordinary_differential_equations423" style="font-size:114%;margin:0 4em"><a href="Numerical_methods_for_ordinary_differential_equations" title="Numerical methods for ordinary differential equations">Numerical methods for ordinary differential equations</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Numerical_methods_for_ordinary_differential_equations#Consistency_and_order" title="Numerical methods for ordinary differential equations">First-order methods</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0;line-height:1.4em; padding:0.33em 0;"><div style="padding:0 0.25em">

<ul><li><a href="Euler_method" title="Euler method">Euler method</a></li>

<li><a href="Backward_Euler_method" title="Backward Euler method">Backward Euler</a></li>

<li><a href="Semi-implicit_Euler_method" title="Semi-implicit Euler method">Semi-implicit Euler</a></li>

<li><a href="Exponential_Euler_method" class="mw-redirect" title="Exponential Euler method">Exponential Euler</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Numerical_methods_for_ordinary_differential_equations#Consistency_and_order" title="Numerical methods for ordinary differential equations">Second-order methods</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0;line-height:1.4em; padding:0.33em 0;"><div style="padding:0 0.25em">

<ul><li><a href="Verlet_integration" title="Verlet integration">Verlet integration</a></li>

<li><a href="Velocity_Verlet" class="mw-redirect" title="Velocity Verlet">Velocity Verlet</a></li>

<li><a href="Trapezoidal_rule_(differential_equations)" title="Trapezoidal rule (differential equations)">Trapezoidal rule</a></li>

<li><a href="Beeman's_algorithm" title="Beeman's algorithm">Beeman's algorithm</a></li>

<li><a href="Midpoint_method" title="Midpoint method">Midpoint method</a></li>

<li><a href="Heun's_method" title="Heun's method">Heun's method</a></li>

<li><a href="Newmark-beta_method" title="Newmark-beta method">Newmark-beta method</a></li>

<li><a href="Leapfrog_integration" title="Leapfrog integration">Leapfrog integration</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Numerical_methods_for_ordinary_differential_equations#Consistency_and_order" title="Numerical methods for ordinary differential equations">Higher-order methods</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0;line-height:1.4em; padding:0.33em 0;"><div style="padding:0 0.25em">

<ul><li><a href="Exponential_integrator" title="Exponential integrator">Exponential integrator</a></li>

<li><a href="Runge%E2%80%93Kutta_methods" title="Runge–Kutta methods">Runge–Kutta methods</a></li>

<li><a href="List_of_Runge%E2%80%93Kutta_methods" title="List of Runge–Kutta methods">List of Runge–Kutta methods</a></li>

<li><a href="General_linear_methods" title="General linear methods">General linear methods</a></li>

<li><a href="Backward_differentiation_formula" title="Backward differentiation formula">Backward differentiation formula</a></li>

<li><a href="Leapfrog_integration#Yoshida_algorithms" title="Leapfrog integration">Yoshida</a></li>

<li><a href="Gauss%E2%80%93Legendre_method" title="Gauss–Legendre method">Gauss–Legendre method</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theory</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0;line-height:1.4em; padding:0.33em 0;"><div style="padding:0 0.25em">

<ul><li><a href="Symplectic_integrator" title="Symplectic integrator">Symplectic integrator</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0;line-height:1.4em; padding:0.33em 0;"><div style="padding:0 0.25em">

<ul><li><a href="Numerical_methods_for_partial_differential_equations" title="Numerical methods for partial differential equations">Numerical methods for partial differential equations</a></li>

<li><a href="Numerical_integration" title="Numerical integration">Numerical integration</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

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