The finite volume method is a method for representing and evaluating partial differential equations as algebraic equations.

Similar to the finite difference method, values are calculated at discrete places on a meshed geometry. "Finite volume" refers to the small volume surrounding each node point on a mesh. In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem.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 conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many computational fluid dynamics packages.

1D example

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Consider the integral form of a simple 1D conservative system defined by the following partial differential equation,

$\backslash rho\_t\; +\; f\_x\backslash left(\backslash rho\; \backslash right)=0$

For a particular cell, we can define the volume average value of $\{\backslash rho\; \backslash left(\; x,\; t\; \backslash right)\}\backslash$ at time $\{t\; =\; t\_1\; \}\backslash$ and $\{\; x\; \backslash in\; \backslash leftx\_1\; ,\; x\_2\; \backslash right\}\backslash$, as,

$\backslash bar\{\backslash rho\}\_i\; \backslash left(\; t\_1\; \backslash right)\; =\; \backslash frac\{1\}\{x\_2\; -\; x\_1\}\; \backslash int\_\{x\_1\}^\{x2\}\; \backslash rho\; \backslash left(x,t\_1\; \backslash right)\; dx$,

and at time $\{t\; =\; t\_2\}\backslash$ as,

$\backslash bar\{\backslash rho\}\_i\; \backslash left(\; t\_2\; \backslash right)\; =\; \backslash frac\{1\}\{x\_2\; -\; x\_1\}\; \backslash int\_\{x\_1\}^\{x2\}\; \backslash rho\; \backslash left(x,t\_2\; \backslash right)\; dx$,

where $x\_1\backslash$ and $x\_2\backslash$ represent the faces of the cell in question.

It therefore follows that,

$\backslash bar\{\backslash rho\}\_i\; \backslash left(\; t\_2\; \backslash right)\; =\; \backslash frac\{1\}\{x\_2\; -\; x\_1\}\; \backslash left[\; \backslash int\_\{x\_1\}^\{x2\}\; \backslash rho\; \backslash left(x,t\_1\; \backslash right)\; dx$

+ \int_{t_1}^{t2} f \left(x_1,t \right) dt + \int_{t_1}^{t2} f \left(x_2,t \right) dt \right].

We can also formulate a semi-discrete numerical scheme for the above problem with cell centres indexed as $i\backslash$, and with cell fluxes defined at the cell edges, as follows

$\backslash frac\{d\; \backslash bar\{\backslash rho\}\_i\}\{d\; t\}\; +\; \backslash frac\{1\}\{\backslash Delta\; x\_i\}\; \backslash left[$

f_{i + \frac{1}{2}} - f_{i - \frac{1}{2}} \right] =0 .

Where values for the edge fluxes can be obtained by interpolation or extrapolation. Extrapolation is used in high resolution schemes where shocks or discontinuities are present in the solution. This scheme is conservative as cell averages change through the edge fluxes!

External links

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• The Finite Volume Method (FVM) - An introduction by Oliver Rübenkönig of Albert Ludwigs University of Freiburg, available under the GFDL.

Partial differential equations | Numerical analysis | Fluid dynamics

Finite-Volumen-Verfahren | Méthode des volumes finis | Metodo dei volumi finiti