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

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<p>

<b>Particle methods</b> is a widely used class of numerical algorithms in scientific computing. Its application ranges from <a href="Computational_fluid_dynamics" title="Computational fluid dynamics">computational fluid dynamics</a> (CFD) over <a href="Molecular_dynamics" title="Molecular dynamics">molecular dynamics</a> (MD) to <a href="Discrete_element_method" title="Discrete element method">discrete element methods</a>.

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

<p>One of the earliest particle methods is <a href="Smoothed_particle_hydrodynamics" class="mw-redirect" title="Smoothed particle hydrodynamics">smoothed particle hydrodynamics</a>, presented in 1977.<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> Libersky <i>et al.</i><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> were the first to apply SPH in solid mechanics. The main drawbacks of SPH are inaccurate results near boundaries and tension instability that was first investigated by Swegle.<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>

</p><p>In the 1990s a new class of particle methods emerged. The <a href="Reproducing_kernel_particle_method" class="mw-redirect" title="Reproducing kernel particle method">reproducing kernel particle method</a><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> (RKPM) emerged, the approximation motivated in part to correct the kernel estimate in SPH: to give accuracy near boundaries, in non-uniform discretizations, and higher-order accuracy in general. Notably, in a parallel development, the <a href="Material_point_method" title="Material point method">Material point methods</a> were developed around the same time<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> which offer similar capabilities. During the 1990s and thereafter several other varieties were developed including those listed below.

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<div class="mw-heading mw-heading2"><h2 id="List_of_methods_and_acronyms">List of methods and acronyms</h2></div>

<p>The following numerical methods are generally considered to fall within the general class of "particle" methods. Acronyms are provided in parentheses.

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<ul><li><a href="Smoothed_particle_hydrodynamics" class="mw-redirect" title="Smoothed particle hydrodynamics">Smoothed particle hydrodynamics</a> (SPH) (1977)</li>

<li><a href="Dissipative_particle_dynamics" title="Dissipative particle dynamics">Dissipative particle dynamics</a> (DPD) (1992)</li>

<li><a href="Reproducing_kernel_particle_method" class="mw-redirect" title="Reproducing kernel particle method">Reproducing kernel particle method</a> (RKPM) (1995)</li>

<li><a href="Moving_particle_semi-implicit" class="mw-redirect" title="Moving particle semi-implicit">Moving particle semi-implicit</a> (MPS)</li>

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

<li>Moving particle finite element method (MPFEM)</li>

<li>Cracking particles method (CPM) (2004)</li>

<li>Immersed particle method (IPM) (2006)</li></ul>

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

<p>The mathematical definition of particle methods captures the structural commonalities of all particle methods.<sup id="cite_ref-pahlke_6-0" class="reference"><a href="#cite_note-pahlke-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> It, therefore, allows for formal reasoning across application domains.

The definition is structured into three parts:

First, the particle method algorithm structure, including structural components, namely data structures, and functions.

Second, the definition of a particle method instance. A particle method instance describes a specific problem or setting, which can be solved or simulated using the particle method algorithm.

Third, the definition of the particle state transition function.

The state transition function describes how a particle method proceeds from the instance to the final state using the data structures and functions from the particle method algorithm.<sup id="cite_ref-pahlke_6-1" class="reference"><a href="#cite_note-pahlke-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p><p>A <b>particle method algorithm</b> is a 7-tuple <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 (P,G,u,f,i,e,{\overset {\circ }{e}})}">

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<annotation encoding="application/x-tex">{\displaystyle (P,G,u,f,i,e,{\overset {\circ }{e}})}</annotation>

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</math></span><img src="./a9727c340916be21221d5f9e91eb4032b26456cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.163ex; height:3.509ex;" alt="{\displaystyle (P,G,u,f,i,e,{\overset {\circ }{e}})}" loading="lazy"></span>, consisting of the two data structures

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<mtext>the particle space,</mtext>

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<mtext>the global variable space,</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&P:=A_{1}\times A_{2}\times ...\times A_{n}&&{\text{the particle space,}}\\&G:=B_{1}\times B_{2}\times ...\times B_{m}&&{\text{the global variable space,}}\end{aligned}}}</annotation>

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</math></span><img src="./2b32c1ffd163bd1c21ea915cf4f93ff41aa8ca08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:54.813ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}&P:=A_{1}\times A_{2}\times ...\times A_{n}&&{\text{the particle space,}}\\&G:=B_{1}\times B_{2}\times ...\times B_{m}&&{\text{the global variable space,}}\end{aligned}}}" loading="lazy"></span>

</p><p>such 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 [G\times P^{*}]}">

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<annotation encoding="application/x-tex">{\displaystyle [G\times P^{*}]}</annotation>

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</math></span><img src="./23bef26dc0c24bd88afdb07866ae0d3e7898bc51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.837ex; height:2.843ex;" alt="{\displaystyle [G\times P^{*}]}" loading="lazy"></span>

is the state space of the particle method, and five functions:

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<mtext>the neighborhood function,</mtext>

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<mtext>the stopping condition,</mtext>

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<mtext>the interact function,</mtext>

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<mtext>the evolve function,</mtext>

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<mtext>the evolve function of the global variable.</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&u:[G\times P^{*}]\times \mathbb {N} \rightarrow \mathbb {N} ^{*}&&{\text{the neighborhood function,}}\\&f:G\rightarrow \{\top ,\bot \}&&{\text{the stopping condition,}}\\&i:G\times P\times P\rightarrow P\times P&&{\text{the interact function,}}\\&e:G\times P\rightarrow G\times P^{*}\ &&{\text{the evolve function,}}\\&{\overset {\circ }{e}}:G\rightarrow G&&{\text{the evolve function of the global variable.}}\end{aligned}}}</annotation>

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</math></span><img src="./0f3d891b0cfcde5e4a0c6e0b350e9831b35279c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.002ex; margin-bottom: -0.336ex; width:70.365ex; height:15.676ex;" alt="{\displaystyle {\begin{aligned}&u:[G\times P^{*}]\times \mathbb {N} \rightarrow \mathbb {N} ^{*}&&{\text{the neighborhood function,}}\\&f:G\rightarrow \{\top ,\bot \}&&{\text{the stopping condition,}}\\&i:G\times P\times P\rightarrow P\times P&&{\text{the interact function,}}\\&e:G\times P\rightarrow G\times P^{*}\ &&{\text{the evolve function,}}\\&{\overset {\circ }{e}}:G\rightarrow G&&{\text{the evolve function of the global variable.}}\end{aligned}}}" loading="lazy"></span>

</p><p>An initial state defines a <b>particle method instance</b> for a given particle method algorithm <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 (P,G,u,f,i,e,{\overset {\circ }{e}})}">

<mover>

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<annotation encoding="application/x-tex">{\displaystyle (P,G,u,f,i,e,{\overset {\circ }{e}})}</annotation>

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</math></span><img src="./a9727c340916be21221d5f9e91eb4032b26456cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.163ex; height:3.509ex;" alt="{\displaystyle (P,G,u,f,i,e,{\overset {\circ }{e}})}" loading="lazy"></span>:

</p><p><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 [g^{1},\mathbf {p} ^{1}]\in [G\times P^{*}].}">

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<annotation encoding="application/x-tex">{\displaystyle [g^{1},\mathbf {p} ^{1}]\in [G\times P^{*}].}</annotation>

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</math></span><img src="./3d39e87b8e60e5bf0b4859702100207dce0005e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.364ex; height:3.176ex;" alt="{\displaystyle [g^{1},\mathbf {p} ^{1}]\in [G\times P^{*}].}" loading="lazy"></span>

</p><p>The instance consists of an initial value for the global variable <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 g^{1}\in G}">

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<annotation encoding="application/x-tex">{\displaystyle g^{1}\in G}</annotation>

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</math></span><img src="./e7abcd1bf5e79febff194f69db42e56ab6cd8392.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.84ex; height:3.009ex;" alt="{\displaystyle g^{1}\in G}" loading="lazy"></span> and an initial tuple of particles <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 \mathbf {p} ^{1}\in P^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {p} ^{1}\in P^{*}}</annotation>

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</math></span><img src="./27baba10ec1c00522a1b47d887e980b48fd62c1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.256ex; height:3.009ex;" alt="{\displaystyle \mathbf {p} ^{1}\in P^{*}}" loading="lazy"></span>.

</p><p>In a specific particle method, the elements of the tuple <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 (P,G,u,f,i,e,{\overset {\circ }{e}})}">

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</math></span><img src="./a9727c340916be21221d5f9e91eb4032b26456cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.163ex; height:3.509ex;" alt="{\displaystyle (P,G,u,f,i,e,{\overset {\circ }{e}})}" loading="lazy"></span> need to be specified. Given a specific starting point defined by an instance <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 [g^{1},\mathbf {p} ^{1}]}">

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<annotation encoding="application/x-tex">{\displaystyle [g^{1},\mathbf {p} ^{1}]}</annotation>

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</math></span><img src="./3a6e76d1347069eedb0f89e0a4ba84e44a6f0d28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.04ex; height:3.176ex;" alt="{\displaystyle [g^{1},\mathbf {p} ^{1}]}" loading="lazy"></span>, the algorithm proceeds in iterations.

Each iteration corresponds to one state transition 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 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> that advances the current state of the particle method <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 [g^{t},\mathbf {p} ^{t}]}">

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<annotation encoding="application/x-tex">{\displaystyle [g^{t},\mathbf {p} ^{t}]}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle [g^{t+1},\mathbf {p} ^{t+1}]}</annotation>

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</math></span><img src="./586d51da19026536d7cebd4e1e6f1088ecd814c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.784ex; height:3.176ex;" alt="{\displaystyle [g^{t+1},\mathbf {p} ^{t+1}]}" loading="lazy"></span>.

The state transition uses the functions

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to determine the next state.

The <b>state transition function</b> <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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</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" 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 true}">

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</math></span><img src="./8a6eed06737c40c0afef0a4e92449a1b2700ec6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.301ex; height:2.009ex;" alt="{\displaystyle true}" loading="lazy"></span>. The so-calculated final state is the result of the state transition function. The state transition function is identical for every particle method.

</p><p>The <b>state transition function</b> is defined as

</p><p><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:[G\times P^{*}]\rightarrow [G\times P^{*}]}">

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<annotation encoding="application/x-tex">{\displaystyle S:[G\times P^{*}]\rightarrow [G\times P^{*}]}</annotation>

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</math></span><img src="./d7c8d502c913649a49cde52447d50e3ed5f02def.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.724ex; height:2.843ex;" alt="{\displaystyle S:[G\times P^{*}]\rightarrow [G\times P^{*}]}" loading="lazy"></span>

</p><p>with

</p><p><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 [g^{T},\mathbf {p} ^{T}]:=S([g^{1},\mathbf {p} ^{1}])}">

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<annotation encoding="application/x-tex">{\displaystyle [g^{T},\mathbf {p} ^{T}]:=S([g^{1},\mathbf {p} ^{1}])}</annotation>

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</math></span><img src="./b4b862268056ae291166d6d0a485a79c9e41edeb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.803ex; height:3.176ex;" alt="{\displaystyle [g^{T},\mathbf {p} ^{T}]:=S([g^{1},\mathbf {p} ^{1}])}" loading="lazy"></span>.

</p><p>The pseudo-code illustrates the particle method state transition function:

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</math></span><img src="./ce6b0539481596f1b4522e2eb8d35f755edd903a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.067ex; height:3.176ex;" alt="{\displaystyle [g,\mathbf {p} ]=[g^{1},\mathbf {p} ^{1}]}" loading="lazy"></span>

2 <b>while</b> <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(g)=false}">

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</semantics>

</math></span><img src="./1a670d175500fe22de0337a7705d18c190c9c779.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.678ex; height:2.843ex;" alt="{\displaystyle f(g)=false}" loading="lazy"></span>

3 <b>for</b> <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 j=1}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./91423fb032c471061948939f4a6811bf463780e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:5.246ex; height:2.509ex;" alt="{\displaystyle j=1}" loading="lazy"></span> <b>to</b> <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 |\mathbf {p} |}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./6412e280b338030b7a5201182d0051f1b5b8bb25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.779ex; height:2.843ex;" alt="{\displaystyle |\mathbf {p} |}" loading="lazy"></span>

4 <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 \mathbf {k} =u([g,\mathbf {p} ],j)}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {k} =u([g,\mathbf {p} ],j)}</annotation>

</semantics>

</math></span><img src="./d5bbd6619394c9f2d3dbb594c360fed07a2b90a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.569ex; height:2.843ex;" alt="{\displaystyle \mathbf {k} =u([g,\mathbf {p} ],j)}" loading="lazy"></span>

5 <b>for</b> <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 l=1}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./8fc60b21200ebf2b338c4fa71b103cb697b02bd3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.954ex; height:2.176ex;" alt="{\displaystyle l=1}" loading="lazy"></span> <b>to</b> <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 |\mathbf {k} |}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./28f15f361169715b1002978c24dd66bef1d7fc8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.705ex; height:2.843ex;" alt="{\displaystyle |\mathbf {k} |}" loading="lazy"></span>

6 <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 (p_{j},p_{k_{j}})=i(g,p_{j},p_{k_{j}})}">

<msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (p_{j},p_{k_{j}})=i(g,p_{j},p_{k_{j}})}</annotation>

</semantics>

</math></span><img src="./083e2116875b6b6fbf7549a2b35924492487e58e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:21.84ex; height:3.176ex;" alt="{\displaystyle (p_{j},p_{k_{j}})=i(g,p_{j},p_{k_{j}})}" loading="lazy"></span>

7 <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 \mathbf {q} =()}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./4de1a439d6b56268a6b908e4fef02fb9d1187cf8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.323ex; height:2.843ex;" alt="{\displaystyle \mathbf {q} =()}" loading="lazy"></span>

8 <b>for</b> <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 j=1}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./91423fb032c471061948939f4a6811bf463780e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:5.246ex; height:2.509ex;" alt="{\displaystyle j=1}" loading="lazy"></span> <b>to</b> <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 |\mathbf {p} |}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./6412e280b338030b7a5201182d0051f1b5b8bb25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.779ex; height:2.843ex;" alt="{\displaystyle |\mathbf {p} |}" loading="lazy"></span>

9 <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 (g,{\overline {\mathbf {q} }})=e(g,p_{j})}">

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (g,{\overline {\mathbf {q} }})=e(g,p_{j})}</annotation>

</semantics>

</math></span><img src="./a50ad3cace6c17942eba7122382f12d2362739d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.713ex; height:3.009ex;" alt="{\displaystyle (g,{\overline {\mathbf {q} }})=e(g,p_{j})}" loading="lazy"></span>

10 <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 \mathbf {q} =\mathbf {q} \circ {\overline {\mathbf {q} }}}">

</mrow>

</mrow>

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c52d9cc03c4cc7a53f877d1467086118de028c16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.658ex; height:2.676ex;" alt="{\displaystyle \mathbf {q} =\mathbf {q} \circ {\overline {\mathbf {q} }}}" loading="lazy"></span>

11 <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 \mathbf {p} =\mathbf {q} }">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d8214b31ac41954d8e147f4543edafd93c9f2415.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.999ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} =\mathbf {q} }" loading="lazy"></span>

12 <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 g={\overset {\circ }{e}}(g)}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g={\overset {\circ }{e}}(g)}</annotation>

</semantics>

</math></span><img src="./ab2412cbf857f7793c5ac30c034a703f7e4232f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.223ex; height:3.509ex;" alt="{\displaystyle g={\overset {\circ }{e}}(g)}" loading="lazy"></span>

13 <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 [g^{T},\mathbf {p} ^{T}]=[g,\mathbf {p} ]}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [g^{T},\mathbf {p} ^{T}]=[g,\mathbf {p} ]}</annotation>

</semantics>

</math></span><img src="./11fd7444e11ab7e622e1c655e1df89820ae46c2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.737ex; height:3.176ex;" alt="{\displaystyle [g^{T},\mathbf {p} ^{T}]=[g,\mathbf {p} ]}" loading="lazy"></span>

</pre>

<p>The fat symbols are tuples, <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 \mathbf {p} ,\mathbf {q} }">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./6a23abdc463ddf71b1b7058a3762b8cda6ff8fb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.935ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} ,\mathbf {q} }" loading="lazy"></span> are particle tuples 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 \mathbf {k} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./9ea699cbc1f843f2e855577d57529430ec33a1ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:2.176ex;" alt="{\displaystyle \mathbf {k} }" loading="lazy"></span> is an index tuple. <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 ()}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d7bc8aa05e1302397bb3e7877e842784991351df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.809ex; height:2.843ex;" alt="{\displaystyle ()}" loading="lazy"></span> is the empty tuple. The operator <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 \circ }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./99add39d2b681e2de7ff62422c32704a05c7ec31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \circ }" loading="lazy"></span> is the <a href="Concatenation" title="Concatenation">concatenation</a> of the particle tuples, e.g. <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 (p_{1},p_{2})\circ (p_{3},p_{4},p_{5})=(p_{1},p_{2},p_{3},p_{4},p_{5})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (p_{1},p_{2})\circ (p_{3},p_{4},p_{5})=(p_{1},p_{2},p_{3},p_{4},p_{5})}</annotation>

</semantics>

</math></span><img src="./356ccc7e16f9b2a1377a4658c1edeacdab9b86f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.195ex; height:2.843ex;" alt="{\displaystyle (p_{1},p_{2})\circ (p_{3},p_{4},p_{5})=(p_{1},p_{2},p_{3},p_{4},p_{5})}" 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 |\mathbf {p} |}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./6412e280b338030b7a5201182d0051f1b5b8bb25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.779ex; height:2.843ex;" alt="{\displaystyle |\mathbf {p} |}" loading="lazy"></span> is the number of elements in the tuple <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 \mathbf {p} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./dd73e3862cb92b016721b8c492eadb4e8a577527.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.485ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} }" loading="lazy"></span>, e.g. <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 |(p_{1},p_{2})|=2}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ccc6b5259ae2e702f5e9bf8988cf30a821cd563f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.845ex; height:2.843ex;" alt="{\displaystyle |(p_{1},p_{2})|=2}" loading="lazy"></span>.

</p>

<ul><li><a href="Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></li>

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

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

<li><a href="Stencil_code" class="mw-redirect" title="Stencil code">Stencil code</a></li>

<li><a href="Meshfree_methods" title="Meshfree methods">Meshfree methods</a></li></ul>

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

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

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

<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Gingold RA, Monaghan JJ (1977). Smoothed particle hydrodynamics – theory and application to non-spherical stars. Mon Not R Astron Soc 181:375–389</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">Libersky, L.D., Petscheck, A.G., Carney, T.C., Hipp, J.R., Allahdadi, F.A. (1993). High Strain Lagrangian Hydrodynamics. <i>Journal of Computational Physics</i>.</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">Swegle, J.W., Hicks, D.L., Attaway, S.W. (1995). Smoothed Particle Hydrodynamics Stability Analysis. <i>Journal of Computational Physics</i>. 116(1), 123-134</span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Liu, W.K., Jun, S., Zhang, Y.F. (1995), Reproducing kernel particle methods, <i>International Journal of Numerical Methods in Fluids</i>. 20, 1081-1106. </span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">D. Sulsky, Z., Chen, H. Schreyer (1994). a Particle Method for History-Dependent Materials. <i>Computer Methods in Applied Mechanics and Engineering</i> (118) 1, 179-196.</span>

</li>

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</style><cite id="CITEREFPahlkeSbalzarini2023" class="citation journal cs1">Pahlke, Johannes; Sbalzarini, Ivo F. (March 2023). <a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FOJCS.2023.3254466">"A Unifying Mathematical Definition of Particle Methods"</a>. <i>IEEE Open Journal of the Computer Society</i>. <b>4</b>: <span class="nowrap">97–</span>108. <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/10.1109%2FOJCS.2023.3254466">10.1109/OJCS.2023.3254466</a></span>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:257480034">257480034</a>.</cite> <i><span typeof="mw:File"><a href="https://creativecommons.org/licenses/by/4.0/" title="creativecommons:by/4.0/" class="external"></a></span> This article incorporates text available under the <a href="https://creativecommons.org/licenses/by/4.0/" class="extiw external" title="creativecommons:by/4.0/">CC BY 4.0</a> license.</i></span>

</li>

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<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

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

<ul><li>Liu MB, Liu GR, Zong Z, AN OVERVIEW ON SMOOTHED PARTICLE HYDRODYNAMICS, INTERNATIONAL JOURNAL OF COMPUTATIONAL METHODS Vol. 5 Issue: 1, 135–188, 2008.</li>

<li>Liu, G.R., Liu, M.B. (2003). <i>Smoothed Particle Hydrodynamics, a meshfree and Particle Method</i>, World Scientific, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>981-238-456-1</bdi>.</li></ul>

</div>

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

<ul><li><a rel="nofollow" class="external text" href="https://www.particleworks.com/technical_column01_en.html">Particle Methods</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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