The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations [LeVeque, 2002; Toro, 1999]. Similar to the finite difference method or finite element 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.
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1D example
Consider a simple 1D advection problem defined by the following partial differential equation
Here,
and at time
where
Integrating equation (1) in time, we have:
where
To obtain the volume average of
We assume that
where
We can therefore derive a semi-discrete numerical scheme for the above problem with cell centres indexed as
where values for the edge fluxes,
This method can also be applied to a 2D situation by considering the north and south faces along with the east and west faces around a node.
General conservation law
We can also consider the general conservation law problem, represented by the following PDE,
Here,
On integrating the first term to get the volume average and applying the divergence theorem to the second, this yields
where
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 schemes 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!