Finite volume method

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.

Consider a simple 1D advection problem defined by the following partial differential equation

Here, ${\displaystyle \rho =\rho \left(x,t\right)\ }$ represents the state variable and ${\displaystyle f=f\left(\rho \left(x,t\right)\right)\ }$ represents the flux or flow of ${\displaystyle \rho \ }$. Conventionally, positive ${\displaystyle f\ }$ represents flow to the right while negative ${\displaystyle f\ }$ 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, ${\displaystyle x\ }$, into finite volumes or cells with cell centres indexed as ${\displaystyle i\ }$. For a particular cell, ${\displaystyle i\ }$, we can define the volume average value of ${\displaystyle {\rho }_{i}\left(t\right)=\rho \left(x,t\right)\ }$ at time ${\displaystyle {t=t_{1}}\ }$ and ${\displaystyle {x\in \left[x_{i-{\frac {1}{2}}},x_{i+{\frac {1}{2}}}\right]}\ }$, as

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