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<title>Split-step method</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Split-step method</span></span>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="Numerical_analysis" title="Numerical analysis">numerical analysis</a>, the <b>split-step</b> <b>Fourier</b> <b>method</b> is a <a href="Pseudo-spectral_method" title="Pseudo-spectral method">pseudo-spectral</a> numerical method used to solve nonlinear <a href="Partial_differential_equation" title="Partial differential equation">partial differential equations</a> like the <a href="Nonlinear_Schr%C3%B6dinger_equation" title="Nonlinear Schrödinger equation">nonlinear Schrödinger equation</a>. The name arises for two reasons. First, the method relies on computing the solution in small steps, and treating the linear and the nonlinear steps separately (see below). Second, it is necessary to <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> back and forth because the linear step is made in the <a href="Frequency_domain" title="Frequency domain">frequency domain</a> while the nonlinear step is made in the <a href="Time_domain" title="Time domain">time domain</a>.

</p><p>An example of usage of this method is in the field of light pulse propagation in optical fibers, where the interaction of linear and nonlinear mechanisms makes it difficult to find general analytical solutions. However, the split-step method provides a numerical solution to the problem. Another application of the split-step method that has been gaining a lot of traction since the 2010s is the simulation of <a href="Kerr_frequency_comb" title="Kerr frequency comb">Kerr frequency comb</a> dynamics in <a href="Optical_microresonators" class="mw-redirect" title="Optical microresonators">optical microresonators</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The relative ease of implementation of the <a href="Lugiato%E2%80%93Lefever_equation" title="Lugiato–Lefever equation">Lugiato–Lefever equation</a> with reasonable numerical cost, along with its success in reproducing experimental spectra as well as predicting <a href="Soliton" title="Soliton">soliton</a> behavior in these microresonators has made the method very popular.

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<div class="mw-heading mw-heading2"><h2 id="Description_of_the_method">Description of the method</h2></div>

<p>Consider, for example, the <a href="Nonlinear_Schr%C3%B6dinger_equation" title="Nonlinear Schrödinger equation">nonlinear Schrödinger equation</a><sup id="cite_ref-NonlinearAgrawal_4-0" class="reference"><a href="#cite_note-NonlinearAgrawal-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

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<annotation encoding="application/x-tex">{\displaystyle {\partial A \over \partial z}=-{i\beta _{2} \over 2}{\partial ^{2}A \over \partial t^{2}}+i\gamma |A|^{2}A=[{\hat {D}}+{\hat {N}}]A,}</annotation>

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</math></span><img src="./91cf35a34937a35299f8bc094d1daa83340b18f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:42.139ex; height:6.009ex;" alt="{\displaystyle {\partial A \over \partial z}=-{i\beta _{2} \over 2}{\partial ^{2}A \over \partial t^{2}}+i\gamma |A|^{2}A=[{\hat {D}}+{\hat {N}}]A,}" loading="lazy"></span></dd></dl>

<p>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 A(t,z)}">

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

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</math></span><img src="./2c43168b8fbc3a81d8a5ac3e388448e4908ee720.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.514ex; height:2.843ex;" alt="{\displaystyle A(t,z)}" loading="lazy"></span> describes the pulse envelope in time <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}">

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

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</math></span><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span> at the spatial position <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}">

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

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</math></span><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"></span>. The equation can be split into a linear part,

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<annotation encoding="application/x-tex">{\displaystyle {\partial A_{D} \over \partial z}=-{i\beta _{2} \over 2}{\partial ^{2}A \over \partial t^{2}}={\hat {D}}A,}</annotation>

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</math></span><img src="./2ac79e07f44801cb0a5403e3ab3f49363414f66c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:26.795ex; height:6.009ex;" alt="{\displaystyle {\partial A_{D} \over \partial z}=-{i\beta _{2} \over 2}{\partial ^{2}A \over \partial t^{2}}={\hat {D}}A,}" loading="lazy"></span></dd></dl>

<p>and a nonlinear part,

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<annotation encoding="application/x-tex">{\displaystyle {\partial A_{N} \over \partial z}=i\gamma |A|^{2}A={\hat {N}}A.}</annotation>

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</math></span><img src="./7f5010963998c7fa2cbe521e5492c5ee17cb42d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.138ex; height:5.509ex;" alt="{\displaystyle {\partial A_{N} \over \partial z}=i\gamma |A|^{2}A={\hat {N}}A.}" loading="lazy"></span></dd></dl>

<p>Both the linear and the nonlinear parts have analytical solutions, but the <a href="Nonlinear_Schr%C3%B6dinger_equation" title="Nonlinear Schrödinger equation">nonlinear Schrödinger equation</a> containing both parts does not have a general analytical solution.

</p><p>However, if only a 'small' step <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 taken along <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}">

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

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</math></span><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"></span>, then the two parts can be treated separately with only a 'small' numerical error. One can therefore first take a small nonlinear step,

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<annotation encoding="application/x-tex">{\displaystyle A_{N}(t,z+h)=\exp \left[i\gamma |A(t,z)|^{2}h\right]A(t,z),}</annotation>

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</math></span><img src="./6874b53b7dba023a9eee81619f8f680953954314.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:40.657ex; height:4.843ex;" alt="{\displaystyle A_{N}(t,z+h)=\exp \left[i\gamma |A(t,z)|^{2}h\right]A(t,z),}" loading="lazy"></span></dd></dl>

<p>using the analytical solution. Note that this ansatz imposes <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(z)|^{2}={\text{const}}.}">

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<annotation encoding="application/x-tex">{\displaystyle |A(z)|^{2}={\text{const}}.}</annotation>

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</math></span><img src="./42da6ca07e0fce758f10656efec654458439a2dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.042ex; height:3.343ex;" alt="{\displaystyle |A(z)|^{2}={\text{const}}.}" loading="lazy"></span> and consequently <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 \gamma \in \mathbb {R} }">

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

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</math></span><img src="./2dd30a1a4bcabc7a62e0faedfd8bea5f664ee191.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.781ex; height:2.676ex;" alt="{\displaystyle \gamma \in \mathbb {R} }" loading="lazy"></span>.

</p><p>The dispersion step has an analytical solution in the <a href="Frequency_domain" title="Frequency domain">frequency domain</a>, so it is first necessary to Fourier transform <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_{N}}">

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

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</math></span><img src="./6c52db369561a56f9e38131013723ee8665f7915.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.435ex; height:2.509ex;" alt="{\displaystyle A_{N}}" loading="lazy"></span> using

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {A}}_{N}(\omega ,z)=\int _{-\infty }^{\infty }A_{N}(t,z)\exp[i(\omega -\omega _{0})t]dt}</annotation>

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</math></span><img src="./74a09ad01666c963044950452fdae76324e13a4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.508ex; height:6.009ex;" alt="{\displaystyle {\tilde {A}}_{N}(\omega ,z)=\int _{-\infty }^{\infty }A_{N}(t,z)\exp[i(\omega -\omega _{0})t]dt}" loading="lazy"></span>,</dd></dl>

<p>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 \omega _{0}}">

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

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</math></span><img src="./9a713d16c489051d4f515e12b1f86061c6be799b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{0}}" loading="lazy"></span> is the center frequency of the pulse.

It can be shown that using the above definition of the <a href="Fourier_transform" title="Fourier transform">Fourier transform</a>, the analytical solution to the linear step, commuted with the frequency domain solution for the nonlinear step, is

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {A}}(\omega ,z+h)=\exp \left[{i\beta _{2} \over 2}(\omega -\omega _{0})^{2}h\right]{\tilde {A}}_{N}(\omega ,z).}</annotation>

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</math></span><img src="./73bf242372afcea460e2253338cb2b70be045bea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.926ex; height:6.176ex;" alt="{\displaystyle {\tilde {A}}(\omega ,z+h)=\exp \left[{i\beta _{2} \over 2}(\omega -\omega _{0})^{2}h\right]{\tilde {A}}_{N}(\omega ,z).}" loading="lazy"></span></dd></dl>

<p>By taking the <a href="Inverse_Fourier_transform" class="mw-redirect" title="Inverse Fourier transform">inverse Fourier transform</a> 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 {\tilde {A}}(\omega ,z+h)}">

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {A}}(\omega ,z+h)}</annotation>

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</math></span><img src="./bbfd86034571a225d347c0aedcde351c3064086c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.332ex; height:3.176ex;" alt="{\displaystyle {\tilde {A}}(\omega ,z+h)}" loading="lazy"></span> one obtains <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\left(t,z+h\right)}">

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</math></span><img src="./84d85b1850d6886d3cc1ba37ee2853183fd1efb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.081ex; height:2.843ex;" alt="{\displaystyle A\left(t,z+h\right)}" loading="lazy"></span>; the pulse has thus been propagated a small step <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 N}</annotation>

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</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> times, the pulse can be propagated over a length 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 Nh}">

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</p><p>The above shows how to use the method to propagate a solution forward in space; however, many physics applications, such as studying the evolution of a wave packet describing a particle, require one to propagate the solution forward in time rather than in space. The non-linear Schrödinger equation, when used to govern the time evolution of a wave function, takes the form

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<annotation encoding="application/x-tex">{\displaystyle i\hbar {\partial \psi \over \partial t}=-{{\hbar }^{2} \over {2m}}{\partial ^{2}\psi \over \partial x^{2}}+\gamma |\psi |^{2}\psi =[{\hat {D}}+{\hat {N}}]\psi ,}</annotation>

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</math></span><img src="./4dcf4049cab61325edb17bb94dcce38eb44f8bf2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:42.326ex; height:6.009ex;" alt="{\displaystyle i\hbar {\partial \psi \over \partial t}=-{{\hbar }^{2} \over {2m}}{\partial ^{2}\psi \over \partial x^{2}}+\gamma |\psi |^{2}\psi =[{\hat {D}}+{\hat {N}}]\psi ,}" loading="lazy"></span></dd></dl>

<p>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 \psi (x,t)}">

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

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</math></span><img src="./710fe5889ab03540462927622f3da789e03271d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.526ex; height:2.843ex;" alt="{\displaystyle \psi (x,t)}" loading="lazy"></span> describes the wave function at position <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 x}">

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

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</math></span><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span>. Note that

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</mrow>

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</mfrac>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\hat {D}}=-{{\hbar }^{2} \over {2m}}{\partial ^{2} \over \partial x^{2}}}</annotation>

</semantics>

</math></span><img src="./d74ac96116232105bf9b53b7e7abb660e5486298.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.408ex; height:6.009ex;" alt="{\displaystyle {\hat {D}}=-{{\hbar }^{2} \over {2m}}{\partial ^{2} \over \partial x^{2}}}" 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 {\hat {N}}=\gamma |\psi |^{2}}">

<mover>

</mover>

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</mrow>

<msup>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\hat {N}}=\gamma |\psi |^{2}}</annotation>

</semantics>

</math></span><img src="./c57824dae9c0bd88a93d5d2b9825103a32d6950b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.285ex; height:3.343ex;" alt="{\displaystyle {\hat {N}}=\gamma |\psi |^{2}}" loading="lazy"></span>, and 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 m}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"></span> is the mass of the particle 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 \hbar }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./de68de3a92517953436c93b5a76461d49160cc41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle \hbar }" loading="lazy"></span> is the <a href="Reduced_Planck_constant" class="mw-redirect" title="Reduced Planck constant">reduced Planck constant</a>.</dd></dl>

<p>The formal solution to this equation is a complex exponential, so we have that

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<annotation encoding="application/x-tex">{\displaystyle \psi (x,t)=e^{-it({\hat {D}}+{\hat {N}})/\hbar }\psi (x,0)}</annotation>

</semantics>

</math></span><img src="./4ab3b8a520a2fab41a432503229dcc08263dbbc5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.352ex; height:3.676ex;" alt="{\displaystyle \psi (x,t)=e^{-it({\hat {D}}+{\hat {N}})/\hbar }\psi (x,0)}" loading="lazy"></span>.</dd></dl>

<p>Since <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 {\hat {D}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./53cfa8853c9c1ae9fa5646c8cabf27d41ace0332.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.843ex;" alt="{\displaystyle {\hat {D}}}" 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 {\hat {N}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1b9b81dbfd5ef1a73800b14a8d2e84a00c59667f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.843ex;" alt="{\displaystyle {\hat {N}}}" loading="lazy"></span> are operators, they do not in general commute. However, the <a href="Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula" title="Baker–Campbell–Hausdorff formula">Baker-Campbell-Hausdorff formula</a> can be applied to show that the error from treating them as if they do will be 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 dt^{2}}">

<msup>

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

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</math></span><img src="./01e2609fa5537d38ebd9f68e47bf96935de8c7a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.11ex; height:2.676ex;" alt="{\displaystyle dt^{2}}" loading="lazy"></span> if we are taking a small but finite time step <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 dt}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ebee76a835701fd1f26047a09855f2ea36bb08fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.055ex; height:2.176ex;" alt="{\displaystyle dt}" loading="lazy"></span>. We therefore can write

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<annotation encoding="application/x-tex">{\displaystyle \psi (x,t+dt)\approx e^{-idt{\hat {D}}/\hbar }e^{-idt{\hat {N}}/\hbar }\psi (x,t)}</annotation>

</semantics>

</math></span><img src="./fd4ee02aa91d84801adbfd6a897151f68d9a2aaf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.588ex; height:3.676ex;" alt="{\displaystyle \psi (x,t+dt)\approx e^{-idt{\hat {D}}/\hbar }e^{-idt{\hat {N}}/\hbar }\psi (x,t)}" loading="lazy"></span>.</dd></dl>

<p>The part of this equation involving <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 {\hat {N}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1b9b81dbfd5ef1a73800b14a8d2e84a00c59667f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.843ex;" alt="{\displaystyle {\hat {N}}}" loading="lazy"></span> can be computed directly using the wave function at time <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span>, but to compute the exponential involving <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 {\hat {D}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./53cfa8853c9c1ae9fa5646c8cabf27d41ace0332.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.843ex;" alt="{\displaystyle {\hat {D}}}" loading="lazy"></span> we use the fact that in frequency space, the partial derivative operator can be converted into a number by substituting <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 ik}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./9b504da4ad46420eddc28989e45c6e35990f97aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.014ex; height:2.176ex;" alt="{\displaystyle ik}" loading="lazy"></span> 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 \partial \over \partial x}">

<mfrac>

</mstyle>

<mrow>

</mrow>

</mfrac>

</mrow>

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

</semantics>

</math></span><img src="./51d6fa129a14ddd4a5265d5b8a36b2d511a87f2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.484ex; height:5.509ex;" alt="{\displaystyle \partial \over \partial x}" loading="lazy"></span>, 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 k}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> is the frequency (or more properly, wave number, as we are dealing with a spatial variable and thus transforming to a space of spatial frequencies—i.e. wave numbers) associated with the Fourier transform of whatever is being operated on. Thus, we take the Fourier transform of

</p>

<msup>

<mover>

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<annotation encoding="application/x-tex">{\displaystyle e^{-idt{\hat {N}}/\hbar }\psi (x,t)}</annotation>

</semantics>

</math></span><img src="./3ecbbbc0ff80255f6d5ff7f9acac823f9766384b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.346ex; height:3.676ex;" alt="{\displaystyle e^{-idt{\hat {N}}/\hbar }\psi (x,t)}" loading="lazy"></span>,</dd></dl>

<p>recover the associated wave number, compute the quantity

</p>

<msup>

<msup>

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</msup>

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

</semantics>

</math></span><img src="./df3c13d1a7d5470bedc28efae5d816a342cd2af3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.025ex; height:3.009ex;" alt="{\displaystyle e^{idtk^{2}}}" loading="lazy"></span>,</dd></dl>

<p>and use it to find the product of the complex exponentials involving <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 {\hat {N}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1b9b81dbfd5ef1a73800b14a8d2e84a00c59667f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.843ex;" alt="{\displaystyle {\hat {N}}}" 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 {\hat {D}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./53cfa8853c9c1ae9fa5646c8cabf27d41ace0332.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.843ex;" alt="{\displaystyle {\hat {D}}}" loading="lazy"></span> in frequency space as below:

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<msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle e^{idtk^{2}}F[e^{-idt{\hat {N}}}\psi (x,t)]}</annotation>

</semantics>

</math></span><img src="./cd6f783577b3acc2e87630c8871470e9f33f035e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.659ex; height:3.676ex;" alt="{\displaystyle e^{idtk^{2}}F[e^{-idt{\hat {N}}}\psi (x,t)]}" loading="lazy"></span>,</dd></dl>

<p>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 F}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span> denotes a Fourier transform. We then inverse Fourier transform this expression to find the final result in physical space, yielding the final expression

</p>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \psi (x,t+dt)=F^{-1}[e^{idtk^{2}}F[e^{-idt{\hat {N}}}\psi (x,t)]]}</annotation>

</semantics>

</math></span><img src="./a0ffff02a9fe5a51f3b4981fcf560e318c537193.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.62ex; height:3.676ex;" alt="{\displaystyle \psi (x,t+dt)=F^{-1}[e^{idtk^{2}}F[e^{-idt{\hat {N}}}\psi (x,t)]]}" loading="lazy"></span>.</dd></dl>

<p>A variation on this method is the symmetrized split-step Fourier method, which takes half a time step using one operator, then takes a full-time step with only the other, and then takes a second half time step again with only the first. This method is an improvement upon the generic split-step Fourier method because its 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 dt^{3}}">

<msup>

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</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle dt^{3}}</annotation>

</semantics>

</math></span><img src="./770612f495aa358b4e14c4049c2ad8b8571ec72f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.11ex; height:2.676ex;" alt="{\displaystyle dt^{3}}" loading="lazy"></span> for a time step <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 dt}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ebee76a835701fd1f26047a09855f2ea36bb08fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.055ex; height:2.176ex;" alt="{\displaystyle dt}" loading="lazy"></span>.

The <a href="Fourier_transform" title="Fourier transform">Fourier transforms</a> of this <a href="Algorithm" title="Algorithm">algorithm</a> can be computed relatively fast using the <i><a href="Fast_Fourier_transform" title="Fast Fourier transform">fast Fourier transform (FFT)</a></i>. The split-step Fourier method can therefore be much faster than typical <a href="Finite_difference_method" title="Finite difference method">finite difference methods</a>.<sup id="cite_ref-Taha1984_5-0" class="reference"><a href="#cite_note-Taha1984-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</p>

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

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</style><cite id="CITEREFErkintaloSylvestreRandleCoen2013" class="citation journal cs1"><a href="Miro_Erkintalo" title="Miro Erkintalo">Erkintalo, Miro</a>; Sylvestre, Thibaut; Randle, Hamish G.; Coen, Stéphane (2013-01-01). "Modeling of octave-spanning Kerr frequency combs using a generalized mean-field Lugiato–Lefever model". <i>Optics Letters</i>. <b>38</b> (1): <span class="nowrap">37–</span>39. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1211.1697">1211.1697</a></span>. <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/2013OptL...38...37C">2013OptL...38...37C</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1364%2FOL.38.000037">10.1364/OL.38.000037</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/1539-4794">1539-4794</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/23282830">23282830</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:7248349">7248349</a>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFMalekiSeidelIlchenkoLiang2011" class="citation journal cs1">Maleki, L.; Seidel, D.; Ilchenko, V. S.; Liang, W.; Savchenkov, A. A.; Matsko, A. B. (2011-08-01). "Mode-locked Kerr frequency combs". <i>Optics Letters</i>. <b>36</b> (15): <span class="nowrap">2845–</span>2847. <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/2011OptL...36.2845M">2011OptL...36.2845M</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1364%2FOL.36.002845">10.1364/OL.36.002845</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/1539-4794">1539-4794</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/21808332">21808332</a>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFHanssonWabnitz2016" class="citation journal cs1">Hansson, Tobias; Wabnitz, Stefan (2016). <a rel="nofollow" class="external text" href="https://iris.unibs.it/bitstream/11379/477683/1/nanoph-2016-0012.pdf">"Dynamics of microresonator frequency comb generation: models and stability"</a> <span class="cs1-format">(PDF)</span>. <i>Nanophotonics</i>. <b>5</b> (2): <span class="nowrap">231–</span>243. <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/2016Nanop...5...12H">2016Nanop...5...12H</a>. <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/10.1515%2Fnanoph-2016-0012">10.1515/nanoph-2016-0012</a></span>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2192-8606">2192-8606</a>.</cite></span>

</li>

<li id="cite_note-NonlinearAgrawal-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-NonlinearAgrawal_4-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFAgrawal2001" class="citation book cs1">Agrawal, Govind P. (2001). <i>Nonlinear Fiber Optics</i> (3rd ed.). San Diego, CA, USA: Academic Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-045143-3</bdi>.</cite></span>

</li>

<cite id="CITEREFT._R._Taha_and_M._J._Ablowitz1984" class="citation journal cs1">T. R. Taha and M. J. Ablowitz (1984). "Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation". <i>J. Comput. Phys</i>. <b>55</b> (2): <span class="nowrap">203–</span>230. <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/1984JCoPh..55..203T">1984JCoPh..55..203T</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0021-9991%2884%2990003-2">10.1016/0021-9991(84)90003-2</a>.</cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_references">External references</h2></div>

<ul><li>Thomas E. Murphy, Software, <a rel="nofollow" class="external free" href="http://www.photonics.umd.edu/software/ssprop/">http://www.photonics.umd.edu/software/ssprop/</a></li>

<li>Andrés A. Rieznik, Software, <a rel="nofollow" class="external free" href="https://github.com/arieznik/freeopticsproject">https://github.com/arieznik/freeopticsproject</a></li>

<li>Prof. G. Agrawal, Software, <a rel="nofollow" class="external free" href="http://www.optics.rochester.edu/workgroups/agrawal/grouphomepage.php?pageid=software">http://www.optics.rochester.edu/workgroups/agrawal/grouphomepage.php?pageid=software</a></li>

<li>Thomas Schreiber, Software, <a rel="nofollow" class="external free" href="http://www.fiberdesk.com">http://www.fiberdesk.com</a></li>

<li>Edward J. Grace, Software, <a rel="nofollow" class="external free" href="http://www.mathworks.com/matlabcentral/fileexchange/24016">http://www.mathworks.com/matlabcentral/fileexchange/24016</a></li></ul>

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</style><div id="Numerical_methods_for_partial_differential_equations284" style="font-size:114%;margin:0 4em"><a href="Numerical_methods_for_partial_differential_equations" title="Numerical methods for partial differential equations">Numerical methods for partial differential equations</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Finite_difference_method" title="Finite difference method">Finite difference</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Parabolic_partial_differential_equation" title="Parabolic partial differential equation">Parabolic</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="FTCS_scheme" title="FTCS scheme">Forward-time central-space</a> (FTCS)</li>

<li><a href="Crank%E2%80%93Nicolson_method" title="Crank–Nicolson method">Crank–Nicolson</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Hyperbolic_partial_differential_equation" title="Hyperbolic partial differential equation">Hyperbolic</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Lax%E2%80%93Friedrichs_method" title="Lax–Friedrichs method">Lax–Friedrichs</a></li>

<li><a href="Lax%E2%80%93Wendroff_method" title="Lax–Wendroff method">Lax–Wendroff</a></li>

<li><a href="MacCormack_method" title="MacCormack method">MacCormack</a></li>

<li><a href="Upwind_scheme" title="Upwind scheme">Upwind</a></li>

<li><a href="Method_of_characteristics" title="Method of characteristics">Method of characteristics</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Others</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Alternating_direction_implicit_method" class="mw-redirect" title="Alternating direction implicit method">Alternating direction-implicit</a> (ADI)</li>

<li><a href="Finite-difference_frequency-domain_method" title="Finite-difference frequency-domain method">Finite-difference frequency-domain</a> (FDFD)</li>

<li><a href="Finite-difference_time-domain_method" title="Finite-difference time-domain method">Finite-difference time-domain</a> (FDTD)</li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Finite_volume_method" title="Finite volume method">Finite volume</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Godunov's_scheme" title="Godunov's scheme">Godunov</a></li>

<li><a href="High-resolution_scheme" title="High-resolution scheme">High-resolution</a></li>

<li><a href="MUSCL_scheme" title="MUSCL scheme">Monotonic upstream-centered</a> (MUSCL)</li>

<li><a href="AUSM" class="mw-redirect" title="AUSM">Advection upstream-splitting</a> (AUSM)</li>

<li><a href="Riemann_solver" title="Riemann solver">Riemann solver</a></li>

<li><a href="ENO_methods" title="ENO methods">Essentially non-oscillatory</a> (ENO)</li>

<li><a href="WENO_methods" title="WENO methods">Weighted essentially non-oscillatory</a> (WENO)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Finite_element_method" title="Finite element method">Finite element</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Extended_finite_element_method" title="Extended finite element method">Extended</a> (XFEM)</li>

<li><a href="Discontinuous_Galerkin_method" title="Discontinuous Galerkin method">Discontinuous Galerkin</a> (DG)</li>

<li><a href="Spectral_element_method" title="Spectral element method">Spectral element</a> (SEM)</li>

<li><a href="Mortar_methods" title="Mortar methods">Mortar</a></li>

<li><a href="Gradient_discretisation_method" title="Gradient discretisation method">Gradient discretisation</a> (GDM)</li>

<li><a href="Loubignac_iteration" title="Loubignac iteration">Loubignac iteration</a></li>

<li><a href="Smoothed_finite_element_method" title="Smoothed finite element method">Smoothed</a> (S-FEM)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Meshfree_methods" title="Meshfree methods">Meshless/Meshfree</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Smoothed-particle_hydrodynamics" title="Smoothed-particle hydrodynamics">Smoothed-particle hydrodynamics</a> (SPH)</li>

<li><a href="Peridynamics" title="Peridynamics">Peridynamics</a> (PD)</li>

<li><a href="Moving_particle_semi-implicit_method" title="Moving particle semi-implicit method">Moving particle semi-implicit method</a> (MPS)</li>

<li><a href="Material_point_method" title="Material point method">Material point method</a> (MPM)</li>

<li><a href="Particle-in-cell" title="Particle-in-cell">Particle-in-cell</a> (PIC)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Domain_decomposition_methods" title="Domain decomposition methods">Domain decomposition</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Schur_complement_method" title="Schur complement method">Schur complement</a></li>

<li><a href="Fictitious_domain_method" title="Fictitious domain method">Fictitious domain</a></li>

<li><a href="Schwarz_alternating_method" title="Schwarz alternating method">Schwarz alternating</a>

<ul><li><a href="Additive_Schwarz_method" title="Additive Schwarz method">additive</a></li>

<li><a href="Abstract_additive_Schwarz_method" title="Abstract additive Schwarz method">abstract additive</a></li></ul></li>

<li><a href="Neumann%E2%80%93Dirichlet_method" title="Neumann–Dirichlet method">Neumann–Dirichlet</a></li>

<li><a href="Neumann%E2%80%93Neumann_methods" title="Neumann–Neumann methods">Neumann–Neumann</a></li>

<li><a href="Poincar%C3%A9%E2%80%93Steklov_operator" title="Poincaré–Steklov operator">Poincaré–Steklov operator</a></li>

<li><a href="Balancing_domain_decomposition_method" title="Balancing domain decomposition method">Balancing</a> (BDD)</li>

<li><a href="BDDC" title="BDDC">Balancing by constraints</a> (BDDC)</li>

<li><a href="FETI" title="FETI">Tearing and interconnect</a> (FETI)</li>

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

<ul><li><a href="Spectral_method" title="Spectral method">Spectral</a></li>

<li><a href="Pseudo-spectral_method" title="Pseudo-spectral method">Pseudospectral</a> (DVR)</li>

<li><a href="Method_of_lines" title="Method of lines">Method of lines</a></li>

<li><a href="Multigrid_method" title="Multigrid method">Multigrid</a></li>

<li><a href="Collocation_method" title="Collocation method">Collocation</a></li>

<li><a href="Level-set_method" title="Level-set method">Level-set</a></li>

<li><a href="Boundary_element_method" title="Boundary element method">Boundary element</a>

<ul><li><a href="Method_of_moments_(electromagnetics)" title="Method of moments (electromagnetics)">Method of moments</a></li></ul></li>

<li><a href="Immersed_boundary_method" title="Immersed boundary method">Immersed boundary</a></li>

<li><a href="Analytic_element_method" title="Analytic element method">Analytic element</a></li>

<li><a href="Isogeometric_analysis" title="Isogeometric analysis">Isogeometric analysis</a></li>

<li><a href="Infinite_difference_method" title="Infinite difference method">Infinite difference method</a></li>

<li><a href="Infinite_element_method" title="Infinite element method">Infinite element method</a></li>

<li><a href="Galerkin_method" title="Galerkin method">Galerkin method</a>

<ul><li><a href="Petrov%E2%80%93Galerkin_method" title="Petrov–Galerkin method">Petrov–Galerkin method</a></li></ul></li>

<li><a href="Validated_numerics" title="Validated numerics">Validated numerics</a></li>

<li><a href="Computer-assisted_proof" title="Computer-assisted proof">Computer-assisted proof</a></li>

<li><a href="Integrable_algorithm" title="Integrable algorithm">Integrable algorithm</a></li>

<li><a href="Method_of_fundamental_solutions" title="Method of fundamental solutions">Method of fundamental solutions</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"><div style="padding:0 0.25em">

<ul><li><a href="Numerical_methods_for_ordinary_differential_equations" title="Numerical methods for ordinary differential equations">Numerical methods for ordinary differential equations</a></li>

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

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