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<title>Finite volume method</title>

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

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</style><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title" style="background:#ccccff;display:block;margin-bottom:0.2em;"><a href="Differential_equation" title="Differential equation">Differential equations</a></th></tr><tr><th class="sidebar-heading" style="background:#ddddff;font-size:105%;display:block;margin-bottom:0.4em;">

Scope</th></tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Fields</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="padding-bottom:0;">

<div class="hlist"><ul><li><a href="Natural_science" title="Natural science">Natural sciences</a></li><li><a href="Engineering" title="Engineering">Engineering</a></li></ul></div></th></tr><tr><td class="sidebar-content hlist" style="padding-bottom:0.6em;">

<ul><li><a href="Astronomy" title="Astronomy">Astronomy</a></li>

<li><a href="Physics" title="Physics">Physics</a></li>

<li><a href="Chemistry" title="Chemistry">Chemistry</a></li>

<li><br><a href="Biology" title="Biology">Biology</a></li>

<li><a href="Geology" title="Geology">Geology</a></li></ul></td>

</tr><tr><th class="sidebar-heading" style="padding-bottom:0;">

<a href="Applied_mathematics" title="Applied mathematics">Applied mathematics</a></th></tr><tr><td class="sidebar-content hlist" style="padding-bottom:0.6em;">

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

<li><a href="Chaos_theory" title="Chaos theory">Chaos theory</a></li>

<li><a href="Dynamical_systems" class="mw-redirect" title="Dynamical systems">Dynamical systems</a></li></ul></td>

</tr><tr><th class="sidebar-heading" style="padding-bottom:0;">

<a href="Social_science" title="Social science">Social sciences</a></th></tr><tr><td class="sidebar-content hlist" style="padding-bottom:0.6em;;padding-bottom:0;">

<ul><li><a href="Economics" title="Economics">Economics</a></li>

<li><a href="Population_dynamics" title="Population dynamics">Population dynamics</a></li></ul></td>

</tr></tbody></table>

<hr>

<a href="List_of_named_differential_equations" title="List of named differential equations">List of named differential equations</a></div></div></td>

</tr><tr><th class="sidebar-heading" style="background:#ddddff;font-size:105%;display:block;margin-bottom:0.4em;;display:block;margin-top:0.1em;">

Classification</th></tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Types</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content">

<ul><li><a href="Ordinary_differential_equation" title="Ordinary differential equation">Ordinary</a></li>

<li><a href="Partial_differential_equation" title="Partial differential equation">Partial</a></li>

<li><a href="Differential-algebraic_system_of_equations" title="Differential-algebraic system of equations">Differential-algebraic</a></li>

<li><a href="Integro-differential_equation" title="Integro-differential equation">Integro-differential</a></li>

<li><a href="Fractional_differential_equations" class="mw-redirect" title="Fractional differential equations">Fractional</a></li>

<li><a href="Linear_differential_equation" title="Linear differential equation">Linear</a></li>

<li><a href="Non-linear_differential_equation" class="mw-redirect" title="Non-linear differential equation">Non-linear</a></li></ul>

</div></td>

</tr><tr><th class="sidebar-heading">

By variable type</th></tr><tr><td class="sidebar-content">

<ul><li><a href="Dependent_and_independent_variables" title="Dependent and independent variables">Dependent and independent variables</a></li></ul>

<ul><li><a href="Autonomous_differential_equation" class="mw-redirect" title="Autonomous differential equation">Autonomous</a></li>

<li>Coupled / Decoupled</li>

<li><a href="Exact_differential_equation" title="Exact differential equation">Exact</a></li>

<li><a href="Homogeneous_differential_equation" title="Homogeneous differential equation">Homogeneous</a> / <a href="Non-homogeneous_differential_equation" class="mw-redirect" title="Non-homogeneous differential equation">Nonhomogeneous</a></li></ul>

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Features</th></tr><tr><td class="sidebar-content">

<ul><li><a href="Ordinary_differential_equation#Definitions" title="Ordinary differential equation">Order</a></li>

<li><a href="Differential_operator" title="Differential operator">Operator</a></li></ul>

</div>

<ul><li><a href="Notation_for_differentiation" title="Notation for differentiation">Notation</a></li></ul></td>

</tr></tbody></table></div></div></td>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Relation to processes</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;">

<ul><li><a href="Difference_equation" class="mw-redirect" title="Difference equation">Difference <span style="font-size: 85%;">(discrete analogue)</span></a></li></ul>

<ul><li><a href="Stochastic_differential_equation" title="Stochastic differential equation">Stochastic</a>

<ul><li><a href="Stochastic_partial_differential_equation" title="Stochastic partial differential equation">Stochastic partial</a></li></ul></li>

<li><a href="Delay_differential_equation" title="Delay differential equation">Delay</a></li></ul>

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Solution</th></tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Existence and uniqueness</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;">

<ul><li><a href="Picard%E2%80%93Lindel%C3%B6f_theorem" title="Picard–Lindelöf theorem">Picard–Lindelöf theorem </a></li>

<li><a href="Peano_existence_theorem" title="Peano existence theorem">Peano existence theorem</a></li>

<li><a href="Carath%C3%A9odory's_existence_theorem" title="Carathéodory's existence theorem">Carathéodory's existence theorem</a></li>

<li><a href="Cauchy%E2%80%93Kowalevski_theorem" class="mw-redirect" title="Cauchy–Kowalevski theorem">Cauchy–Kowalevski theorem</a></li></ul></div></div></td>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">General topics</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><div class="hlist">

<ul><li><a href="Initial_condition" title="Initial condition">Initial conditions</a></li>

<li><a href="Boundary_value_problem" title="Boundary value problem">Boundary values</a>

<ul><li><a href="Dirichlet_boundary_condition" title="Dirichlet boundary condition">Dirichlet</a></li>

<li><a href="Neumann_boundary_condition" title="Neumann boundary condition">Neumann</a></li>

<li><a href="Robin_boundary_condition" title="Robin boundary condition">Robin</a></li>

<li><a href="Cauchy_problem" title="Cauchy problem">Cauchy problem</a></li></ul></li>

<li><a href="Wronskian" title="Wronskian">Wronskian</a></li>

<li><a href="Phase_portrait" title="Phase portrait">Phase portrait</a></li>

<li><a href="Lyapunov_stability" title="Lyapunov stability">Lyapunov</a> / <a href="Asymptotic_stability" class="mw-redirect" title="Asymptotic stability">Asymptotic</a> / <a href="Exponential_stability" title="Exponential stability">Exponential stability</a></li>

<li><a href="Rate_of_convergence" title="Rate of convergence">Rate of convergence</a></li>

<li><span class="nowrap"><a href="Power_series_solution_of_differential_equations" title="Power series solution of differential equations">Series</a> / Integral solutions</span></li>

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

<li><a href="Dirac_delta_function" title="Dirac delta function">Dirac delta function</a></li></ul>

</div></div></div></td>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Solution methods</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><div class="hlist">

<ul><li>Inspection</li>

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

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

<li><a href="Exponential_response_formula" title="Exponential response formula">Exponential response formula</a></li>

<li><a href="Finite_difference_method" title="Finite difference method">Finite difference</a> <span style="font-size: 85%;">(<a href="Crank%E2%80%93Nicolson_method" title="Crank–Nicolson method">Crank–Nicolson</a>)</span></li>

<li><a href="Finite_element_method" title="Finite element method">Finite element</a>

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

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

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

<li><a href="Green's_function" title="Green's function">Green's function</a></li>

<li><a href="Integrating_factor" title="Integrating factor">Integrating factor</a></li>

<li><a href="Integral_transform" title="Integral transform">Integral transforms</a></li>

<li><a href="Perturbation_theory" title="Perturbation theory">Perturbation theory</a></li>

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

</div>

<ul><li><a href="Separation_of_variables" title="Separation of variables">Separation of variables</a></li>

<li><a href="Method_of_undetermined_coefficients" title="Method of undetermined coefficients">Undetermined coefficients</a></li>

<li><a href="Variation_of_parameters" title="Variation of parameters">Variation of parameters</a></li></ul></div></div></td>

</tr><tr><th class="sidebar-heading" style="background:#ddddff;font-size:105%;display:block;margin-bottom:0.4em;">

People</th></tr><tr><td class="sidebar-content">

<ul><li><a href="Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li>

<li><a href="Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Gottfried Leibniz</a></li>

<li><a href="Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a></li>

<li><a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li>

<li><a href="Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a></li>

<li><a href="J%C3%B3zef_Maria_Hoene-Wro%C5%84ski" title="Józef Maria Hoene-Wroński">Józef Maria Hoene-Wroński</a></li>

<li><a href="Joseph_Fourier" title="Joseph Fourier">Joseph Fourier</a></li>

<li><a href="Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li>

<li><a href="George_Green_(mathematician)" title="George Green (mathematician)">George Green</a></li>

<li><a href="Carl_David_Tolm%C3%A9_Runge" class="mw-redirect" title="Carl David Tolmé Runge">Carl David Tolmé Runge</a></li>

<li><a href="Martin_Kutta" title="Martin Kutta">Martin Kutta</a></li>

<li><a href="Rudolf_Lipschitz" title="Rudolf Lipschitz">Rudolf Lipschitz</a></li>

<li><a href="Ernst_Lindel%C3%B6f" class="mw-redirect" title="Ernst Lindelöf">Ernst Lindelöf</a></li>

<li><a href="%C3%89mile_Picard" title="Émile Picard">Émile Picard</a></li>

<li><a href="Phyllis_Nicolson" title="Phyllis Nicolson">Phyllis Nicolson</a></li>

<li><a href="John_Crank" title="John Crank">John Crank</a></li></ul>

</div></div></div></td>

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<p>The <b>finite volume method</b> (<b>FVM</b>) is a method for representing and evaluating <a href="Partial_differential_equation" title="Partial differential equation">partial differential equations</a> in the form of algebraic equations.<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>

In the finite volume method, volume integrals in a partial differential equation that contain a <a href="Divergence" title="Divergence">divergence</a> term are converted to <a href="Surface_integral" title="Surface integral">surface integrals</a>, using the <a href="Divergence_theorem" title="Divergence theorem">divergence theorem</a>.

These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are <a href="Conservation_law_(physics)" class="mw-redirect" title="Conservation law (physics)">conservative</a>. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many <a href="Computational_fluid_dynamics" title="Computational fluid dynamics">computational fluid dynamics</a> packages.

"Finite volume" refers to the small volume surrounding each node point on a mesh.<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>

</p><p>Finite volume methods can be compared and contrasted with the <a href="Finite_difference_method" title="Finite difference method">finite difference methods</a>, which approximate derivatives using nodal values, or <a href="Finite_element_method" title="Finite element method">finite element methods</a>, which create local approximations of a solution using local data, and construct a global approximation by stitching them together. In contrast a finite volume method evaluates exact expressions for the <i>average</i> value of the solution over some volume, and uses this data to construct approximations of the solution within cells.<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><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>

</p>

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

<p>Consider a simple 1D <a href="Advection" title="Advection">advection</a> problem:

</p>

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</style><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><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 {\frac {\partial \rho }{\partial t}}+{\frac {\partial f}{\partial x}}=0,\quad t\geq 0.}">

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \rho }{\partial t}}+{\frac {\partial f}{\partial x}}=0,\quad t\geq 0.}</annotation>

</semantics>

</math></span><img src="./5057d5a558dfef7cb7770bb9006b7b24a6d630f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.045ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \rho }{\partial t}}+{\frac {\partial f}{\partial x}}=0,\quad t\geq 0.}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span></td></tr></tbody></table>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1bce67bbbde532c45187eafb1b200725a43b4c72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.902ex; height:2.843ex;" alt="{\displaystyle \rho =\rho \left(x,t\right)}" loading="lazy"></span> represents the state variable and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=f\left(\rho \left(x,t\right)\right)}">

<mrow>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./f5d6ca630a4f086cf7d98d4e7291963832f0025a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.454ex; height:2.843ex;" alt="{\displaystyle f=f\left(\rho \left(x,t\right)\right)}" loading="lazy"></span> represents the <a href="Flux" title="Flux">flux</a> or flow 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 \rho }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"></span>. Conventionally, positive <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="./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> represents flow to the right while negative <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="./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> represents flow to the left. If we assume that equation (<b><a href="#math_1">1</a></b>) represents a flowing medium of constant area, we can sub-divide the spatial domain, <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span>, into <i>finite volumes</i> or <i>cells</i> with cell centers indexed as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span>. For a particular cell, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span>, we can define the <i>volume average</i> value of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rho }_{i}\left(t\right)=\rho \left(x,t\right)}">

<msub>

</mrow>

</mrow>

</msub>

<mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1afd6ce20e0325de64f0514dc70f561977e9a004.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.738ex; height:2.843ex;" alt="{\displaystyle {\rho }_{i}\left(t\right)=\rho \left(x,t\right)}" loading="lazy"></span> 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=t_{1}}}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./012cb3ad8f7ab1d14d2990145611560c5956e58e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.832ex; height:2.343ex;" alt="{\displaystyle {t=t_{1}}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {x\in \left[x_{i-1/2},x_{i+1/2}\right]}}">

<mrow>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {x\in \left[x_{i-1/2},x_{i+1/2}\right]}}</annotation>

</semantics>

</math></span><img src="./87ba289e228b3272e83206ab38c923fe52017705.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.891ex; height:3.343ex;" alt="{\displaystyle {x\in \left[x_{i-1/2},x_{i+1/2}\right]}}" loading="lazy"></span>, as

</p>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</msubsup>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {\rho }}_{i}\left(t_{1}\right)={\frac {1}{x_{i+1/2}-x_{i-1/2}}}\int _{x_{i-1/2}}^{x_{i+1/2}}\rho \left(x,t_{1}\right)\,dx,}</annotation>

</semantics>

</math></span><img src="./36d1732ae9993c796aa00aa5a983072371a4760e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:44.486ex; height:6.676ex;" alt="{\displaystyle {\bar {\rho }}_{i}\left(t_{1}\right)={\frac {1}{x_{i+1/2}-x_{i-1/2}}}\int _{x_{i-1/2}}^{x_{i+1/2}}\rho \left(x,t_{1}\right)\,dx,}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span></td></tr></tbody></table>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./402c7ec02c034a37c33e571e3ae6e53d6bccfc5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.832ex; height:2.343ex;" alt="{\displaystyle t=t_{2}}" loading="lazy"></span> as,

</p>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</msubsup>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {\rho }}_{i}\left(t_{2}\right)={\frac {1}{x_{i+1/2}-x_{i-1/2}}}\int _{x_{i-1/2}}^{x_{i+1/2}}\rho \left(x,t_{2}\right)\,dx,}</annotation>

</semantics>

</math></span><img src="./66243e49dbfb3c12d7e48ec3055bb2ce31aa956a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:44.486ex; height:6.676ex;" alt="{\displaystyle {\bar {\rho }}_{i}\left(t_{2}\right)={\frac {1}{x_{i+1/2}-x_{i-1/2}}}\int _{x_{i-1/2}}^{x_{i+1/2}}\rho \left(x,t_{2}\right)\,dx,}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_3" class="reference nourlexpansion" style="font-weight:bold;">3</span></td></tr></tbody></table>

<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 x_{i-1/2}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./a3590b77630830aacb299edcf05742d83cb8425c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.874ex; height:2.509ex;" alt="{\displaystyle x_{i-1/2}}" 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 x_{i+1/2}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i+1/2}}</annotation>

</semantics>

</math></span><img src="./f716b7f266f1adfcff8147225a6e09d5fcc2a984.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.874ex; height:2.509ex;" alt="{\displaystyle x_{i+1/2}}" loading="lazy"></span> represent locations of the upstream and downstream faces or edges respectively of the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{\text{th}}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./76cc827ec109594f9da6862138d76775cd733866.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.588ex; height:2.676ex;" alt="{\displaystyle i^{\text{th}}}" loading="lazy"></span> cell.

</p><p>Integrating equation (<b><a href="#math_1">1</a></b>) in time, we have:

</p>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho \left(x,t_{2}\right)=\rho \left(x,t_{1}\right)-\int _{t_{1}}^{t_{2}}f_{x}\left(x,t\right)\,dt,}</annotation>

</semantics>

</math></span><img src="./7467791291c33b0be3cdc6fa6f86657eb5dc9e02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.947ex; height:6.509ex;" alt="{\displaystyle \rho \left(x,t_{2}\right)=\rho \left(x,t_{1}\right)-\int _{t_{1}}^{t_{2}}f_{x}\left(x,t\right)\,dt,}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_4" class="reference nourlexpansion" style="font-weight:bold;">4</span></td></tr></tbody></table>

<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_{x}={\frac {\partial f}{\partial x}}}">

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./9f8cf93d59499aa3f1187e27ba92ce263e63ece8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.894ex; height:5.676ex;" alt="{\displaystyle f_{x}={\frac {\partial f}{\partial x}}}" loading="lazy"></span>.

</p><p>To obtain the volume average 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 \rho \left(x,t\right)}">

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./2095ee06478d558f8a9f64101db4dcd7b74b4697.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.602ex; height:2.843ex;" alt="{\displaystyle \rho \left(x,t\right)}" loading="lazy"></span> 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=t_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./402c7ec02c034a37c33e571e3ae6e53d6bccfc5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.832ex; height:2.343ex;" alt="{\displaystyle t=t_{2}}" loading="lazy"></span>, we integrate <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho \left(x,t_{2}\right)}">

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho \left(x,t_{2}\right)}</annotation>

</semantics>

</math></span><img src="./3b76003971feba6125a00858a76ab02579e45231.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.656ex; height:2.843ex;" alt="{\displaystyle \rho \left(x,t_{2}\right)}" loading="lazy"></span> over the cell volume, <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 \left[x_{i-1/2},x_{i+1/2}\right]}">

<mrow>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left[x_{i-1/2},x_{i+1/2}\right]}</annotation>

</semantics>

</math></span><img src="./bb2ed58fccc5748f30731e4c59ef34ca5cea1d57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.721ex; height:3.343ex;" alt="{\displaystyle \left[x_{i-1/2},x_{i+1/2}\right]}" loading="lazy"></span> and divide the result by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x_{i}=x_{i+1/2}-x_{i-1/2}}">

<msub>

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</msub>

<msub>

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</msub>

<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta x_{i}=x_{i+1/2}-x_{i-1/2}}</annotation>

</semantics>

</math></span><img src="./718d80fb199eb46adf7eca3f0e3a1c7dbe4d0fe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:21.752ex; height:3.009ex;" alt="{\displaystyle \Delta x_{i}=x_{i+1/2}-x_{i-1/2}}" loading="lazy"></span>, i.e.

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<annotation encoding="application/x-tex">{\displaystyle {\bar {\rho }}_{i}\left(t_{2}\right)={\frac {1}{\Delta x_{i}}}\int _{x_{i-1/2}}^{x_{i+1/2}}\left\{\rho \left(x,t_{1}\right)-\int _{t_{1}}^{t_{2}}f_{x}\left(x,t\right)dt\right\}dx.}</annotation>

</semantics>

</math></span><img src="./3432066918519b3e5d99084cedf71bfc835361a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:54.566ex; height:7.009ex;" alt="{\displaystyle {\bar {\rho }}_{i}\left(t_{2}\right)={\frac {1}{\Delta x_{i}}}\int _{x_{i-1/2}}^{x_{i+1/2}}\left\{\rho \left(x,t_{1}\right)-\int _{t_{1}}^{t_{2}}f_{x}\left(x,t\right)dt\right\}dx.}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_5" class="reference nourlexpansion" style="font-weight:bold;">5</span></td></tr></tbody></table>

<p>We assume 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 f\ }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./6ba5b1577da8634aff7f6868a9c1c9a1d83c37ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.859ex; height:2.509ex;" alt="{\displaystyle f\ }" loading="lazy"></span> is well behaved and that we can reverse the order of integration. Also, recall that flow is normal to the unit area of the cell. Now, since in one dimension <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_{x}\triangleq \nabla \cdot f}">

<msub>

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</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f_{x}\triangleq \nabla \cdot f}</annotation>

</semantics>

</math></span><img src="./4268b4741c88816c8ff23fb80b96f1bbcfafe31c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.304ex; height:2.843ex;" alt="{\displaystyle f_{x}\triangleq \nabla \cdot f}" loading="lazy"></span>, we can apply the <a href="Divergence_theorem" title="Divergence theorem">divergence theorem</a>, i.e. <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 \oint _{v}\nabla \cdot fdv=\oint _{S}f\,dS}">

<msub>

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<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \oint _{v}\nabla \cdot fdv=\oint _{S}f\,dS}</annotation>

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</math></span><img src="./47d8f0ad97387703d2549136bdd2ff0ee7617749.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.398ex; height:5.676ex;" alt="{\displaystyle \oint _{v}\nabla \cdot fdv=\oint _{S}f\,dS}" loading="lazy"></span>, and substitute for the volume integral of the <a href="Divergence" title="Divergence">divergence</a> with the values of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"></span> evaluated at the cell surface (edges <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_{i-1/2}}">

<msub>

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</msub>

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

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</math></span><img src="./a3590b77630830aacb299edcf05742d83cb8425c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.874ex; height:2.509ex;" alt="{\displaystyle x_{i-1/2}}" 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 x_{i+1/2}}">

<msub>

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</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i+1/2}}</annotation>

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</math></span><img src="./f716b7f266f1adfcff8147225a6e09d5fcc2a984.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.874ex; height:2.509ex;" alt="{\displaystyle x_{i+1/2}}" loading="lazy"></span>) of the finite volume as follows:

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\bar {\rho }}_{i}\left(t_{2}\right)={\bar {\rho }}_{i}\left(t_{1}\right)-{\frac {1}{\Delta x_{i}}}\left(\int _{t_{1}}^{t_{2}}f_{i+1/2}dt-\int _{t_{1}}^{t_{2}}f_{i-1/2}dt\right).}</annotation>

</semantics>

</math></span><img src="./74a1b0024c3c7b05f016e72485ade23a36437b7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:55.549ex; height:6.509ex;" alt="{\displaystyle {\bar {\rho }}_{i}\left(t_{2}\right)={\bar {\rho }}_{i}\left(t_{1}\right)-{\frac {1}{\Delta x_{i}}}\left(\int _{t_{1}}^{t_{2}}f_{i+1/2}dt-\int _{t_{1}}^{t_{2}}f_{i-1/2}dt\right).}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_6" class="reference nourlexpansion" style="font-weight:bold;">6</span></td></tr></tbody></table>

<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_{i\pm 1/2}=f\left(x_{i\pm 1/2},t\right)}">

<msub>

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</msub>

<mrow>

<mrow>

<msub>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{i\pm 1/2}=f\left(x_{i\pm 1/2},t\right)}</annotation>

</semantics>

</math></span><img src="./b04880a81bc78c46f659a8aa51c101c1aa1941a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.325ex; height:3.343ex;" alt="{\displaystyle f_{i\pm 1/2}=f\left(x_{i\pm 1/2},t\right)}" loading="lazy"></span>.

</p><p>We can therefore derive a <i>semi-discrete</i> numerical scheme for the above problem with cell centers indexed as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span>, and with cell edge fluxes indexed as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\pm 1/2}">

</mrow>

</mstyle>

</mrow>

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

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</math></span><img src="./03b8d6b4f49d303f04ac399a219e7e0435f8ce64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.13ex; height:2.843ex;" alt="{\displaystyle i\pm 1/2}" loading="lazy"></span>, by differentiating (<b><a href="#math_6">6</a></b>) with respect to time to obtain:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d{\bar {\rho }}_{i}}{dt}}+{\frac {1}{\Delta x_{i}}}\left[f_{i+1/2}-f_{i-1/2}\right]=0,}</annotation>

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</math></span><img src="./a429ba99f6c9f91eaef96820da38d5c29aa42713.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.411ex; height:6.009ex;" alt="{\displaystyle {\frac {d{\bar {\rho }}_{i}}{dt}}+{\frac {1}{\Delta x_{i}}}\left[f_{i+1/2}-f_{i-1/2}\right]=0,}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_7" class="reference nourlexpansion" style="font-weight:bold;">7</span></td></tr></tbody></table>

<p>where values for the edge fluxes, <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_{i\pm 1/2}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle f_{i\pm 1/2}}</annotation>

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</math></span><img src="./e9038603d420551e1fec2f4eb31f3ccdc2aaf4e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.683ex; height:3.009ex;" alt="{\displaystyle f_{i\pm 1/2}}" loading="lazy"></span>, can be reconstructed by <a href="Interpolation" title="Interpolation">interpolation</a> or <a href="Extrapolation" title="Extrapolation">extrapolation</a> of the cell averages. Equation (<b><a href="#math_7">7</a></b>) is <i>exact</i> for the volume averages; i.e., no approximations have been made during its derivation.

</p><p>This method can also be applied to a <a href="Finite_volume_method_for_two_dimensional_diffusion_problem" title="Finite volume method for two dimensional diffusion problem">2D</a> situation by considering the north and south faces along with the east and west faces around a node.

</p>

<div class="mw-heading mw-heading2"><h2 id="General_conservation_law">General conservation law</h2></div>

<p>We can also consider the general <a href="Conservation_law_(physics)" class="mw-redirect" title="Conservation law (physics)">conservation law</a> problem, represented by the following <a href="Partial_differential_equation" title="Partial differential equation">PDE</a>,

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}+\nabla \cdot {\mathbf {f} }\left({\mathbf {u} }\right)={\mathbf {0} }.}</annotation>

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</math></span><img src="./b51c972d2e2acbf84a58fa1ee23f9d35c202cd98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.912ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}+\nabla \cdot {\mathbf {f} }\left({\mathbf {u} }\right)={\mathbf {0} }.}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_8" class="reference nourlexpansion" style="font-weight:bold;">8</span></td></tr></tbody></table>

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

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</math></span><img src="./261e20fe101de02a771021d9d4466c0ad3e352d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} }" loading="lazy"></span> represents a vector of states 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 {f} }">

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</mstyle>

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

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</math></span><img src="./dc6194e680a4e7c521f2178c50eea302843a852d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.053ex; height:2.176ex;" alt="{\displaystyle \mathbf {f} }" loading="lazy"></span> represents the corresponding <a href="Flux" title="Flux">flux</a> tensor. Again we can sub-divide the spatial domain into finite volumes or cells. For a particular cell, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

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

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</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span>, we take the volume integral over the total volume of the cell, <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 v_{i}}">

<msub>

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

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</math></span><img src="./7dffe5726650f6daac54829972a94f38eb8ec127.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.927ex; height:2.009ex;" alt="{\displaystyle v_{i}}" loading="lazy"></span>, which gives,

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<annotation encoding="application/x-tex">{\displaystyle \int _{v_{i}}{\frac {\partial \mathbf {u} }{\partial t}}\,dv+\int _{v_{i}}\nabla \cdot {\mathbf {f} }\left({\mathbf {u} }\right)\,dv={\mathbf {0} }.}</annotation>

</semantics>

</math></span><img src="./ed639ec68a35e88dbb5105a73b722a9b7e82acbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:32.428ex; height:6.176ex;" alt="{\displaystyle \int _{v_{i}}{\frac {\partial \mathbf {u} }{\partial t}}\,dv+\int _{v_{i}}\nabla \cdot {\mathbf {f} }\left({\mathbf {u} }\right)\,dv={\mathbf {0} }.}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_9" class="reference nourlexpansion" style="font-weight:bold;">9</span></td></tr></tbody></table>

<p>On integrating the first term to get the <i>volume average</i> and applying the <i>divergence theorem</i> to the second, this yields

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle v_{i}{{d{\mathbf {\bar {u}} }_{i}} \over dt}+\oint _{S_{i}}{\mathbf {f} }\left({\mathbf {u} }\right)\cdot {\mathbf {n} }\ dS={\mathbf {0} },}</annotation>

</semantics>

</math></span><img src="./c090c1c39237f8daf0f9fdb4e5b2996b4711942c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.926ex; height:6.176ex;" alt="{\displaystyle v_{i}{{d{\mathbf {\bar {u}} }_{i}} \over dt}+\oint _{S_{i}}{\mathbf {f} }\left({\mathbf {u} }\right)\cdot {\mathbf {n} }\ dS={\mathbf {0} },}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_10" class="reference nourlexpansion" style="font-weight:bold;">10</span></td></tr></tbody></table>

<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 S_{i}}">

<msub>

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

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</math></span><img src="./de6e810a93f67802ecb603ee0e3324005c6e583e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.225ex; height:2.509ex;" alt="{\displaystyle S_{i}}" loading="lazy"></span> represents the total surface area of the cell 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 {n} }}">

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</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./fc886abe4692651edd93479b9ef47f9cea584788.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle {\mathbf {n} }}" loading="lazy"></span> is a unit vector normal to the surface and pointing outward. So, finally, we are able to present the general result equivalent to (<b><a href="#math_8">8</a></b>), i.e.

</p>

<mfrac>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {{d{\mathbf {\bar {u}} }_{i}} \over {dt}}+{{1} \over {v_{i}}}\oint _{S_{i}}{\mathbf {f} }\left({\mathbf {u} }\right)\cdot {\mathbf {n} }\ dS={\mathbf {0} }.}</annotation>

</semantics>

</math></span><img src="./d5a2d21cf21dee1659f80e8ae72cda4b9fbb8bc7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.149ex; height:6.176ex;" alt="{\displaystyle {{d{\mathbf {\bar {u}} }_{i}} \over {dt}}+{{1} \over {v_{i}}}\oint _{S_{i}}{\mathbf {f} }\left({\mathbf {u} }\right)\cdot {\mathbf {n} }\ dS={\mathbf {0} }.}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_11" class="reference nourlexpansion" style="font-weight:bold;">11</span></td></tr></tbody></table>

<p>Again, values for the edge fluxes can be reconstructed by interpolation or extrapolation of the cell averages. The actual numerical scheme will depend upon problem geometry and mesh construction. <a href="MUSCL_scheme" title="MUSCL scheme">MUSCL</a> reconstruction is often used in <a href="High_resolution_scheme" class="mw-redirect" title="High resolution scheme">high resolution schemes</a> where shocks or discontinuities are present in the solution.

</p><p>Finite volume schemes are conservative as cell averages change through the edge fluxes. In other words, <i>one cell's loss is always another cell's gain</i>!

</p>

<ul><li><a href="Finite_element_method" title="Finite element method">Finite element method</a></li>

<li><a href="Flux_limiter" title="Flux limiter">Flux limiter</a></li>

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

<li><a href="Godunov's_theorem" title="Godunov's theorem">Godunov's theorem</a></li>

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

<li><a href="KIVA_(software)" title="KIVA (software)">KIVA (software)</a></li>

<li><a href="MIT_General_Circulation_Model" title="MIT General Circulation Model">MIT General Circulation Model</a></li>

<li><a href="MUSCL_scheme" title="MUSCL scheme">MUSCL scheme</a></li>

<li><a href="Sergei_K._Godunov" class="mw-redirect" title="Sergei K. Godunov">Sergei K. Godunov</a></li>

<li><a href="Total_variation_diminishing" title="Total variation diminishing">Total variation diminishing</a></li>

<li><a href="Finite_volume_method_for_unsteady_flow" title="Finite volume method for unsteady flow">Finite volume method for unsteady flow</a></li></ul>

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

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

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</style><cite id="CITEREFLeVeque2002" class="citation book cs1">LeVeque, Randall (2002). <a rel="nofollow" class="external text" href="https://www.cambridge.org/core/books/finite-volume-methods-for-hyperbolic-problems/97D5D1ACB1926DA1D4D52EAD6909E2B9"><i>Finite Volume Methods for Hyperbolic Problems</i></a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780511791253</bdi>.</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="CITEREFWantaSmolikKryszynWróblewski2021" class="citation journal cs1">Wanta, D.; Smolik, W. T.; Kryszyn, J.; Wróblewski, P.; Midura, M. (October 2021). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs40010-021-00748-7">"A Finite Volume Method using a Quadtree Non-Uniform Structured Mesh for Modeling in Electrical Capacitance Tomography"</a>. <i>Proceedings of the National Academy of Sciences, India Section A: Physical Sciences</i>. <b>92</b> (3): <span class="nowrap">443–</span>452. <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/10.1007%2Fs40010-021-00748-7">10.1007/s40010-021-00748-7</a></span>.</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="CITEREFFallahBaileyCrossTaylor2000" class="citation journal cs1">Fallah, N. A.; Bailey, C.; Cross, M.; Taylor, G. A. (2000-06-01). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0307-904X%2899%2900047-5">"Comparison of finite element and finite volume methods application in geometrically nonlinear stress analysis"</a>. <i>Applied Mathematical Modelling</i>. <b>24</b> (7): <span class="nowrap">439–</span>455. <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/10.1016%2FS0307-904X%2899%2900047-5">10.1016/S0307-904X(99)00047-5</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/0307-904X">0307-904X</a>.</cite></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"><cite id="CITEREFRanganayakulu,_C._(Chennu)2018" class="citation book cs1">Ranganayakulu, C. (Chennu) (2 February 2018). "Chapter 3, Section 3.1". <i>Compact heat exchangers : analysis, design and optimization using FEM and CFD approach</i>. Seetharamu, K. N. Hoboken, NJ. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-119-42435-2</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1006524487">1006524487</a>.</cite><span class="cs1-maint citation-comment"><code class="cs1-code">{{cite book}}</code>: CS1 maint: location missing publisher (link)</span></span>

</li>

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

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

<ul><li>Eymard, R. Gallouët, T. R., <a href="Rapha%C3%A8le_Herbin" title="Raphaèle Herbin">Herbin, R.</a> (2000) <i>The finite volume method</i> Handbook of Numerical Analysis, Vol. VII, 2000, p. 713–1020. Editors: P.G. Ciarlet and J.L. Lions.</li>

<li>Hirsch, C. (1990), <i>Numerical Computation of Internal and External Flows, Volume 2: Computational Methods for Inviscid and Viscous Flows</i>, Wiley.</li>

<li>Laney, Culbert B. (1998), <i>Computational Gas Dynamics</i>, Cambridge University Press.</li>

<li>LeVeque, Randall (1990), <i>Numerical Methods for Conservation Laws</i>, ETH Lectures in Mathematics Series, Birkhauser-Verlag.</li>

<li>LeVeque, Randall (2002), <i>Finite Volume Methods for Hyperbolic Problems</i>, Cambridge University Press.</li>

<li>Patankar, Suhas V. (1980), <i>Numerical Heat Transfer and Fluid Flow</i>, Hemisphere.</li>

<li>Tannehill, John C., et al., (1997), <i>Computational Fluid mechanics and Heat Transfer</i>, 2nd Ed., Taylor and Francis.</li>

<li>Toro, E. F. (1999), <i>Riemann Solvers and Numerical Methods for Fluid Dynamics</i>, Springer-Verlag.</li>

<li>Wesseling, Pieter (2001), <i>Principles of Computational Fluid Dynamics</i>, Springer-Verlag.</li></ul>

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

<ul><li><a rel="nofollow" class="external text" href="https://hal.science/hal-02100732v2/file/bookevol.pdf">Finite volume methods</a> by R. Eymard, T Gallouët and <a href="Rapha%C3%A8le_Herbin" title="Raphaèle Herbin">R. Herbin</a>, update of the article published in Handbook of Numerical Analysis, 2000</li>

<li><cite id="CITEREFRübenkönig" class="citation web cs1">Rübenkönig, Oliver. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20091002233707/http://www.imtek.uni-freiburg.de/simulation/mathematica/IMSweb/imsTOC/Lectures%20and%20Tips/Simulation%20I/FVM_introDocu.html">"The Finite Volume Method (FVM) – An introduction"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.imtek.uni-freiburg.de/simulation/mathematica/IMSweb/imsTOC/Lectures%20and%20Tips/Simulation%20I/FVM_introDocu.html">the original</a> on 2009-10-02.</cite>, available under the <a href="GNU_Free_Document_License" class="mw-redirect" title="GNU Free Document License">GFDL</a>.</li>

<li><a rel="nofollow" class="external text" href="http://www.ctcms.nist.gov/fipy/">FiPy: A Finite Volume PDE Solver Using Python</a> from NIST.</li>

<li><a rel="nofollow" class="external text" href="http://depts.washington.edu/clawpack/">CLAWPACK</a>: a software package designed to compute numerical solutions to hyperbolic partial differential equations using a wave propagation approach</li></ul>

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</style></div><div role="navigation" class="navbox" aria-labelledby="Numerical_methods_for_partial_differential_equations284" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><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%"></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>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Differential_equations71" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Differential_equations71" style="font-size:114%;margin:0 4em"><a href="Differential_equation" title="Differential equation">Differential equations</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Classification</th><td class="navbox-list-with-group navbox-list navbox-odd" 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%">Operations</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="Differential_operator" title="Differential operator">Differential operator</a></li>

<li><a href="Notation_for_differentiation" title="Notation for differentiation">Notation for differentiation</a></li>

<li><a href="Ordinary_differential_equation" title="Ordinary differential equation">Ordinary</a></li>

<li><a href="Partial_differential_equation" title="Partial differential equation">Partial</a></li>

<li><a href="Differential-algebraic_system_of_equations" title="Differential-algebraic system of equations">Differential-algebraic</a></li>

<li><a href="Integro-differential_equation" title="Integro-differential equation">Integro-differential</a></li>

<li><a href="Fractional_differential_equations" class="mw-redirect" title="Fractional differential equations">Fractional</a></li>

<li><a href="Linear_differential_equation" title="Linear differential equation">Linear</a></li>

<li><a href="Non-linear_differential_equation" class="mw-redirect" title="Non-linear differential equation">Non-linear</a></li>

<li><a href="Holonomic_function" title="Holonomic function">Holonomic</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Attributes of variables</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="Dependent_and_independent_variables" title="Dependent and independent variables">Dependent and independent variables</a></li>

<li><a href="Homogeneous_differential_equation" title="Homogeneous differential equation">Homogeneous</a></li>

<li><a href="Non-homogeneous_differential_equation" class="mw-redirect" title="Non-homogeneous differential equation">Nonhomogeneous</a></li>

<li><a href="Differential_equation" title="Differential equation">Coupled</a></li>

<li><a href="Differential_equation" title="Differential equation">Decoupled</a></li>

<li><a href="Differential_equation" title="Differential equation">Order</a></li>

<li><a href="Differential_equation" title="Differential equation">Degree</a></li>

<li><a href="Autonomous_system_(mathematics)" title="Autonomous system (mathematics)">Autonomous</a></li>

<li><a href="Exact_differential_equation" title="Exact differential equation">Exact differential equation</a></li>

<li><a href="Jet_bundle#Partial_differential_equations" title="Jet bundle">On jet bundles</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Relation to processes</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="Difference_equation" class="mw-redirect" title="Difference equation">Difference</a> (discrete analogue)</li>

<li><a href="Stochastic_differential_equation" title="Stochastic differential equation">Stochastic</a>

<ul><li><a href="Stochastic_partial_differential_equation" title="Stochastic partial differential equation">Stochastic partial</a></li></ul></li>

<li><a href="Delay_differential_equation" title="Delay differential equation">Delay</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Solutions</th><td class="navbox-list-with-group navbox-list navbox-odd" 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%">Existence/uniqueness</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="Picard%E2%80%93Lindel%C3%B6f_theorem" title="Picard–Lindelöf theorem">Picard–Lindelöf theorem</a></li>

<li><a href="Peano_existence_theorem" title="Peano existence theorem">Peano existence theorem</a></li>

<li><a href="Carath%C3%A9odory's_existence_theorem" title="Carathéodory's existence theorem">Carathéodory's existence theorem</a></li>

<li><a href="Cauchy%E2%80%93Kowalevski_theorem" class="mw-redirect" title="Cauchy–Kowalevski theorem">Cauchy–Kowalevski theorem</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Solution topics</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="Wronskian" title="Wronskian">Wronskian</a></li>

<li><a href="Phase_portrait" title="Phase portrait">Phase portrait</a></li>

<li><a href="Phase_space" title="Phase space">Phase space</a></li>

<li><a href="Lyapunov_stability" title="Lyapunov stability">Lyapunov stability</a></li>

<li><a href="Asymptotic_stability" class="mw-redirect" title="Asymptotic stability">Asymptotic stability</a></li>

<li><a href="Exponential_stability" title="Exponential stability">Exponential stability</a></li>

<li><a href="Rate_of_convergence" title="Rate of convergence">Rate of convergence</a></li>

<li><a href="Power_series_solution_of_differential_equations" title="Power series solution of differential equations">Series solutions</a></li>

<li><a href="Integral" title="Integral">Integral</a> solutions</li>

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

<li><a href="Dirac_delta_function" title="Dirac delta function">Dirac delta function</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Solution methods</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="List_of_mathematical_jargon" class="mw-redirect" title="List of mathematical jargon">Inspection</a></li>

<li><a href="Integration_by_substitution" title="Integration by substitution">Substitution</a></li>

<li><a href="Separation_of_variables" title="Separation of variables">Separation of variables</a></li>

<li><a href="Method_of_undetermined_coefficients" title="Method of undetermined coefficients">Method of undetermined coefficients</a></li>

<li><a href="Variation_of_parameters" title="Variation of parameters">Variation of parameters</a></li>

<li><a href="Integrating_factor" title="Integrating factor">Integrating factor</a></li>

<li><a href="Integral_transform" title="Integral transform">Integral transforms</a></li>

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

<li><a href="Finite_difference_method" title="Finite difference method">Finite difference method</a></li>

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

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

<li><a href="Finite_element_method" title="Finite element method">Finite element method</a></li>

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

<li><a href="Perturbation_theory" title="Perturbation theory">Perturbation theory</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Examples</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="List_of_named_differential_equations" title="List of named differential equations">List of named differential equations</a></li>

<li><a href="List_of_linear_ordinary_differential_equations" title="List of linear ordinary differential equations">List of linear ordinary differential equations</a></li>

<li><a href="List_of_nonlinear_ordinary_differential_equations" title="List of nonlinear ordinary differential equations">List of nonlinear ordinary differential equations</a></li>

<li><a href="List_of_nonlinear_partial_differential_equations" title="List of nonlinear partial differential equations">List of nonlinear partial differential equations</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mathematicians</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="Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li>

<li><a href="Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a></li>

<li><a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li>

<li><a href="Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a></li>

<li><a href="%C3%89mile_Picard" title="Émile Picard">Émile Picard</a></li>

<li><a href="J%C3%B3zef_Maria_Hoene-Wro%C5%84ski" title="Józef Maria Hoene-Wroński">Józef Maria Hoene-Wroński</a></li>

<li><a href="Ernst_Leonard_Lindel%C3%B6f" title="Ernst Leonard Lindelöf">Ernst Lindelöf</a></li>

<li><a href="Rudolf_Lipschitz" title="Rudolf Lipschitz">Rudolf Lipschitz</a></li>

<li><a href="Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a></li>

<li><a href="Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li>

<li><a href="John_Crank" title="John Crank">John Crank</a></li>

<li><a href="Phyllis_Nicolson" title="Phyllis Nicolson">Phyllis Nicolson</a></li>

<li><a href="Carl_David_Tolm%C3%A9_Runge" class="mw-redirect" title="Carl David Tolmé Runge">Carl David Tolmé Runge</a></li>

<li><a href="Martin_Kutta" title="Martin Kutta">Martin Kutta</a></li>

<li><a href="Sofya_Kovalevskaya" title="Sofya Kovalevskaya">Sofya Kovalevskaya</a></li></ul>

</div></td></tr></tbody></table></div>

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</style></div><div role="navigation" class="navbox authority-control" aria-labelledby="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q1401936#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata1493" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q1401936#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata1493" style="font-size:114%;margin:0 4em">Authority control databases </div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">National</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Finite-Volumen-Methode"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4220855-5">Germany</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Finite volume method"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh95001595">United States</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Volumes finis, Méthodes de"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb150249492">France</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Volumes finis, Méthodes de"><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb150249492">BnF data</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007537003305171">Israel</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://lux.collections.yale.edu/view/concept/4f3293a8-4893-484b-90a2-58ccd6acc8c0">Yale LUX</a></span></li></ul></div></td></tr></tbody></table></div></div><div><div class="zim-footer">

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