Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Finite volume method
open-in-new
<h2 id="example">Example</h2> <p>Consider a simple 1D <a href="/facts/Advection/3rn50V2r">advection</a> problem: </p> <table><tbody><tr><td> ∂ ρ ∂ t + ∂ f ∂ x = 0 , t ≥ 0. {\displaystyle {\frac {\partial \rho }{\partial t}}+{\frac {\partial f}{\partial x}}=0,\quad t\geq 0.} </td> <td></td> <td>1</td></tr></tbody></table> <p>Here, ρ = ρ ( x , t ) {\displaystyle \rho =\rho \left(x,t\right)} represents the state variable and f = f ( ρ ( x , t ) ) {\displaystyle f=f\left(\rho \left(x,t\right)\right)} represents the <a href="/facts/Flux/Aw3P5Bvb">flux</a> or flow of ρ {\displaystyle \rho } . Conventionally, positive f {\displaystyle f} represents flow to the right while negative f {\displaystyle f} represents flow to the left. If we assume that equation (1) represents a flowing medium of constant area, we can sub-divide the spatial domain, x {\displaystyle x} , into <i>finite volumes</i> or <i>cells</i> with cell centers indexed as i {\displaystyle i} . For a particular cell, i {\displaystyle i} , we can define the <i>volume average</i> value of ρ i ( t ) = ρ ( x , t ) {\displaystyle {\rho }_{i}\left(t\right)=\rho \left(x,t\right)} at time t = t 1 {\displaystyle {t=t_{1}}} and x ∈ [ x i − 1 / 2 , x i + 1 / 2 ] {\displaystyle {x\in \left[x_{i-1/2},x_{i+1/2}\right]}} , as </p> <table><tbody><tr><td> ρ ¯ i ( t 1 ) = 1 x i + 1 / 2 − x i − 1 / 2 ∫ x i − 1 / 2 x i + 1 / 2 ρ ( x , t 1 ) d x , {\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,} </td> <td></td> <td>2</td></tr></tbody></table> <p>and at time t = t 2 {\displaystyle t=t_{2}} as, </p> <table><tbody><tr><td> ρ ¯ i ( t 2 ) = 1 x i + 1 / 2 − x i − 1 / 2 ∫ x i − 1 / 2 x i + 1 / 2 ρ ( x , t 2 ) d x , {\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,} </td> <td></td> <td>3</td></tr></tbody></table> <p>where x i − 1 / 2 {\displaystyle x_{i-1/2}} and x i + 1 / 2 {\displaystyle x_{i+1/2}} represent locations of the upstream and downstream faces or edges respectively of the i th {\displaystyle i^{\text{th}}} cell. </p><p>Integrating equation (1) in time, we have: </p> <table><tbody><tr><td> ρ ( x , t 2 ) = ρ ( x , t 1 ) − ∫ t 1 t 2 f x ( x , t ) d t , {\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,} </td> <td></td> <td>4</td></tr></tbody></table> <p>where f x = ∂ f ∂ x {\displaystyle f_{x}={\frac {\partial f}{\partial x}}} . </p><p>To obtain the volume average of ρ ( x , t ) {\displaystyle \rho \left(x,t\right)} at time t = t 2 {\displaystyle t=t_{2}} , we integrate ρ ( x , t 2 ) {\displaystyle \rho \left(x,t_{2}\right)} over the cell volume, [ x i − 1 / 2 , x i + 1 / 2 ] {\displaystyle \left[x_{i-1/2},x_{i+1/2}\right]} and divide the result by Δ x i = x i + 1 / 2 − x i − 1 / 2 {\displaystyle \Delta x_{i}=x_{i+1/2}-x_{i-1/2}} , i.e. </p> <table><tbody><tr><td> ρ ¯ i ( t 2 ) = 1 Δ x i ∫ x i − 1 / 2 x i + 1 / 2 { ρ ( x , t 1 ) − ∫ t 1 t 2 f x ( x , t ) d t } d x . {\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.} </td> <td></td> <td>5</td></tr></tbody></table> <p>We assume that f {\displaystyle f\ } 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 f x ≜ ∇ ⋅ f {\displaystyle f_{x}\triangleq \nabla \cdot f} , we can apply the <a href="/facts/Divergence_theorem/pUGKF1N2">divergence theorem</a>, i.e. ∮ v ∇ ⋅ f d v = ∮ S f d S {\displaystyle \oint _{v}\nabla \cdot fdv=\oint _{S}f\,dS} , and substitute for the volume integral of the <a href="/facts/Divergence/zpwwkFoU">divergence</a> with the values of f ( x ) {\displaystyle f(x)} evaluated at the cell surface (edges x i − 1 / 2 {\displaystyle x_{i-1/2}} and x i + 1 / 2 {\displaystyle x_{i+1/2}} ) of the finite volume as follows: </p> <table><tbody><tr><td> ρ ¯ i ( t 2 ) = ρ ¯ i ( t 1 ) − 1 Δ x i ( ∫ t 1 t 2 f i + 1 / 2 d t − ∫ t 1 t 2 f i − 1 / 2 d t ) . {\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).} </td> <td></td> <td>6</td></tr></tbody></table> <p>where f i ± 1 / 2 = f ( x i ± 1 / 2 , t ) {\displaystyle f_{i\pm 1/2}=f\left(x_{i\pm 1/2},t\right)} . </p><p>We can therefore derive a <i>semi-discrete</i> numerical scheme for the above problem with cell centers indexed as i {\displaystyle i} , and with cell edge fluxes indexed as i ± 1 / 2 {\displaystyle i\pm 1/2} , by differentiating (6) with respect to time to obtain: </p> <table><tbody><tr><td> d ρ ¯ i d t + 1 Δ x i [ f i + 1 / 2 − f i − 1 / 2 ] = 0 , {\displaystyle {\frac {d{\bar {\rho }}_{i}}{dt}}+{\frac {1}{\Delta x_{i}}}\left[f_{i+1/2}-f_{i-1/2}\right]=0,} </td> <td></td> <td>7</td></tr></tbody></table> <p>where values for the edge fluxes, f i ± 1 / 2 {\displaystyle f_{i\pm 1/2}} , can be reconstructed by <a href="/facts/Interpolation/8PyetX4f">interpolation</a> or <a href="/facts/Extrapolation/wggUrqjF">extrapolation</a> of the cell averages. Equation (7) 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="/facts/Finite_volume_method_for_two_dimensional_diffusion_problem/8Xmwzgae">2D</a> situation by considering the north and south faces along with the east and west faces around a node. </p> <h2 id="general-conservation-law">General conservation law</h2> <p>We can also consider the general <a href="/facts/Conservation_law_(physics)/6g1Ay2FR">conservation law</a> problem, represented by the following <a href="/facts/Partial_differential_equation/DyPsBlv7">PDE</a>, </p> <table><tbody><tr><td> ∂ u ∂ t + ∇ ⋅ f ( u ) = 0 . {\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}+\nabla \cdot {\mathbf {f} }\left({\mathbf {u} }\right)={\mathbf {0} }.} </td> <td></td> <td>8</td></tr></tbody></table> <p>Here, u {\displaystyle \mathbf {u} } represents a vector of states and f {\displaystyle \mathbf {f} } represents the corresponding <a href="/facts/Flux/Aw3P5Bvb">flux</a> tensor. Again we can sub-divide the spatial domain into finite volumes or cells. For a particular cell, i {\displaystyle i} , we take the volume integral over the total volume of the cell, v i {\displaystyle v_{i}} , which gives, </p> <table><tbody><tr><td> ∫ v i ∂ u ∂ t d v + ∫ v i ∇ ⋅ f ( u ) d v = 0 . {\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} }.} </td> <td></td> <td>9</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 </p> <table><tbody><tr><td> v i d u ¯ i d t + ∮ S i f ( u ) ⋅ n d S = 0 , {\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} },} </td> <td></td> <td>10</td></tr></tbody></table> <p>where S i {\displaystyle S_{i}} represents the total surface area of the cell and n {\displaystyle {\mathbf {n} }} is a unit vector normal to the surface and pointing outward. So, finally, we are able to present the general result equivalent to (8), i.e. </p> <table><tbody><tr><td> d u ¯ i d t + 1 v i ∮ S i f ( u ) ⋅ n d S = 0 . {\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} }.} </td> <td></td> <td>11</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="/facts/MUSCL_scheme/bIBaWqyW">MUSCL</a> reconstruction is often used in <a href="/facts/High_resolution_scheme/N56RRHnS">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> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Finite_element_method/eUcfP1vn">Finite element method</a></li> <li><a href="/facts/Flux_limiter/vWBzcJx0">Flux limiter</a></li> <li><a href="/facts/Godunov%2527s_scheme/l4YaCvoG">Godunov's scheme</a></li> <li><a href="/facts/Godunov%2527s_theorem/CoLFQRm7">Godunov's theorem</a></li> <li><a href="/facts/High-resolution_scheme/N56RRHnS">High-resolution scheme</a></li> <li><a href="/facts/KIVA_(software)/398MWA4p">KIVA (software)</a></li> <li><a href="/facts/MIT_General_Circulation_Model/B0x78c64">MIT General Circulation Model</a></li> <li><a href="/facts/MUSCL_scheme/bIBaWqyW">MUSCL scheme</a></li> <li><a href="/facts/Sergei_K._Godunov/IaYktIaI">Sergei K. Godunov</a></li> <li><a href="/facts/Total_variation_diminishing/BxKKPAkR">Total variation diminishing</a></li> <li><a href="/facts/Finite_volume_method_for_unsteady_flow/mn5XPUpE">Finite volume method for unsteady flow</a></li></ul> <h2 id="further-reading">Further reading</h2> <ul><li>Eymard, R. Gallouët, T. R., <a href="/facts/Rapha%25C3%25A8le_Herbin/bV7fSSQG">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> <h2 id="external-links">External links</h2> <ul><li><a href="https://hal.science/hal-02100732v2/file/bookevol.pdf">Finite volume methods</a> by R. Eymard, T Gallouët and <a href="/facts/Rapha%25C3%25A8le_Herbin/bV7fSSQG">R. Herbin</a>, update of the article published in Handbook of Numerical Analysis, 2000</li> <li>Rübenkönig, Oliver. <a 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 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., available under the <a href="/facts/GNU_Free_Document_License/IhVHF3zP">GFDL</a>.</li> <li><a href="http://www.ctcms.nist.gov/fipy/">FiPy: A Finite Volume PDE Solver Using Python</a> from NIST.</li> <li><a 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> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>LeVeque, Randall (2002). Finite Volume Methods for Hyperbolic Problems. ISBN 9780511791253. <a href="9780511791253" target="_blank">9780511791253</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Wanta, D.; Smolik, W. T.; Kryszyn, J.; Wróblewski, P.; Midura, M. (October 2021). "A Finite Volume Method using a Quadtree Non-Uniform Structured Mesh for Modeling in Electrical Capacitance Tomography". Proceedings of the National Academy of Sciences, India Section A: Physical Sciences. 92 (3): 443–452. doi:10.1007/s40010-021-00748-7. <a href="https://doi.org/10.1007%2Fs40010-021-00748-7" target="_blank">https://doi.org/10.1007%2Fs40010-021-00748-7</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Fallah, N. A.; Bailey, C.; Cross, M.; Taylor, G. A. (2000-06-01). "Comparison of finite element and finite volume methods application in geometrically nonlinear stress analysis". Applied Mathematical Modelling. 24 (7): 439–455. doi:10.1016/S0307-904X(99)00047-5. ISSN 0307-904X. <a href="https://doi.org/10.1016%2FS0307-904X%2899%2900047-5" target="_blank">https://doi.org/10.1016%2FS0307-904X%2899%2900047-5</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Ranganayakulu, C. (Chennu) (2 February 2018). "Chapter 3, Section 3.1". Compact heat exchangers : analysis, design and optimization using FEM and CFD approach. Seetharamu, K. N. Hoboken, NJ. ISBN 978-1-119-42435-2. OCLC 1006524487.{{cite book}}: CS1 maint: location missing publisher (link) <a href="978-1-119-42435-2" target="_blank">978-1-119-42435-2</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>