# Finite volume method

The

**finite volume method**is a method for representing and evaluatingpartial differential equation s as algebraic equations. Similar to thefinite 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 adivergence term are converted tosurface integral s, using thedivergence 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 manycomputational fluid dynamics packages.**1D example**Consider a simple 1D

advection problem defined by the followingpartial differential equation :$quad\; (1)\; qquad\; qquad\; frac\{partial\; ho\}\{partial\; t\}+frac\{partial\; f\}\{partial\; x\}=0,quad\; tge0.$

Here, $ho=\; ho\; left(\; x,t\; ight)$ represents the state variable and $f=f\; left(\; ho\; left(\; x,t\; ight)\; ight)$ represents the

flux or flow of $ho$. Conventionally, positive $f$ represents flow to the right whilst negative $f$ 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, $x$, into "finite volumes" or "cells" with cell centres indexed as $i$. For a particular cell, $i$, we can define the "volume average" value of $\{\; ho\; \}\_i\; left(\; t\; ight)\; =\; ho\; left(\; x,\; t\; ight)$ at time $\{t\; =\; t\_1\; \}$ and $\{\; x\; in\; left\; [\; x\_\{i-frac\{1\}\{2\; ,\; x\_\{i+frac\{1\}\{2\; ight]\; \}$, as:$quad\; (2)\; qquad\; qquad\; ar\{\; ho\}\_i\; left(\; t\_1\; ight)\; =\; frac\{1\}\{\; x\_\{i+frac\{1\}\{2\; -\; x\_\{i-frac\{1\}\{2\}\; int\_\{x\_\{i-frac\{1\}\{2\}^\{x\_\{i+frac\{1\}\{2\}\; ho\; left(x,t\_1\; ight),\; dx\; ,$

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

:$quad\; (3)\; qquad\; qquad\; ar\{\; ho\}\_i\; left(\; t\_2\; ight)\; =\; frac\{1\}\{x\_\{i+frac\{1\}\{2\; -\; x\_\{i-frac\{1\}\{2\}\; int\_\{x\_\{i-frac\{1\}\{2\}^\{x\_\{i+frac\{1\}\{2\}\; ho\; left(x,t\_2\; ight),\; dx\; ,$

where $x\_\{i-frac\{1\}\{2$ and $x\_\{i+frac\{1\}\{2$ represent locations of the upstream and downstream faces or edges respectively of the $i^\{th\}$ cell.

Integrating equation (1) in time, we have:

:$quad\; (4)\; qquad\; qquad\; ho\; left(\; x,\; t\_2\; ight)\; =\; ho\; left(\; x,\; t\_1\; ight)\; +\; int\_\{t\_1\}^\{t\_2\}\; f\_x\; left(\; ho\; left(\; x,t\; ight)\; ight),\; dt.$

To obtain the volume average of $holeft(x,t\; ight)$ at time $t=t\_\{2\}$, we integrate $holeft(x,t\_2\; ight)$ over the cell volume, $v\_i$ and divide the result by $v\_i$, i.e.

:$quad\; (5)\; qquad\; qquad\; ar\{\; ho\}\_\{i\}left(\; t\_\{2\}\; ight)\; =frac\{1\}\{v\_i\}int\_\{v\_i\}left\{\; holeft(\; x,t\_\{1\}\; ight)\; +int\_\{t\_\{1^\{t\_2\}f\_\{x\}left(\; ho\; left(\; x,t\; ight)\; ight)\; dt\; ight\}\; dv.$

We assume that $f$ 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 $f\_x\; riangleq\; abla\; f$, we can apply the

divergence theorem and substitute for the volume integral of thedivergence with the values of $f(x)$ at the cell edges $x\_\{i-frac\{1\}\{2$ and $x\_\{i+frac\{1\}\{2$ of the finite volume as follows::$quad\; (6)\; qquad\; qquad\; ar\{\; ho\}\_i\; left(\; t\_2\; ight)\; =\; frac\{1\}\{Delta\; x\_\{i\; left\; [\; int\_\{x\_\{i-frac\{1\}\{2\}^\{x\_\{i+frac\{1\}\{2\}\; ho\; left(x,t\_1\; ight),\; dx\; +\; int\_\{t\_1\}^\{t\_2\}\; f\_\{i\; +\; frac\{1\}\{2\; dt-\; int\_\{t\_1\}^\{t\_2\}\; f\_\{i\; -\; frac\{1\}\{2\; dt\; ight]\; .$

where $Delta\; x\_i\; =\; x\_\{i+frac\{1\}\{2-x\_\{i-frac\{1\}\{2$ and $f\_\{i\; pm\; frac\{1\}\{2\; =f\; left(\; ho\; left(\; x\_\{i\; pm\; frac\{1\}\{2,\; t\; ight)\; ight)$.

We can therefore derive a "semi-discrete" numerical scheme for the above problem with cell centres indexed as $i$, and with cell edge fluxes indexed as $ipmfrac\{1\}\{2\}$, by differentiating (6) with respect to time to obtain:

:$quad\; (7)\; qquad\; qquad\; frac\{d\; ar\{\; ho\}\_i\}\{d\; t\}\; +\; frac\{1\}\{Delta\; x\_i\}\; left\; [\; f\_\{i\; +\; frac\{1\}\{2\; -\; f\_\{i\; -\; frac\{1\}\{2\; ight]\; =0\; ,$

where values for the edge fluxes, $f\_\{i\; pm\; frac\{1\}\{2$, can be reconstructed by interpolation or extrapolation of the cell averages. It should be noted that equation (7) is "exact" for the volume averages; i.e., no approximations have been made during its derivation.

**General hyperbolic problem**We can also consider a general hyperbolic problem, represented by the following PDE,

:$quad\; (8)\; qquad\; qquad$partial {mathbf u over {partial t + abla cdot {mathbf f}left( {mathbf u } ight) = {mathbf 0} .

Here, $\{mathbf\; u\}$ represents a vector of states and $mathbf\; f$ represents the corresponding

flux vector. Again we can sub-divide the spatial domain into finite volumes or cells. For a particular cell, $i$ , we take the volume integral over the total volume of the cell, $v\; \_\{i\}$, which gives,:$quad\; (9)\; qquad\; qquad\; int\; \_\{v\_\{i$partial {mathbf u over {partial t, dv + int _{v_{i abla cdot {mathbf f}left( {mathbf u } ight), dv = {mathbf 0} .

On integrating the first term to get the "volume average" and applying the "divergence theorem" to the second, this yields

:$quad\; (10)\; qquad\; qquad\; v\_\{i\}$d {mathbf {ar u} }_{i} } over {dt + oint _{S_{i} } {mathbf f} left( {mathbf u } ight) dS = {mathbf 0},

where $S\_\{i\}$ represents the total surface area of the cell. So, finally, we are able to present the general result equivalent to (7), i.e.

:$quad\; (11)\; qquad\; qquad$d {mathbf {ar u} }_{i} } over {dt + 1} over {v_{i } oint _{S_{i} } {mathbf f} left( {mathbf u } ight) dS = {mathbf 0} .

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. MUSCL reconstruction is often used in

high resolution scheme s where shocks or discontinuities are present in the solution.Finite volume schemes are conservative as cell averages change through the edge fluxes. In other words, "one cell's loss is another cell's gain"!

**ee also***

Flux limiter

*Godunov's theorem

*High-resolution scheme

*MUSCL scheme

*Sergei K. Godunov

*Total variation diminishing

*Finite element method **References****External links*** [

*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*] by Oliver Rübenkönig ofAlbert Ludwigs University of Freiburg , available under the GFDL.

*Wikimedia Foundation.
2010.*

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