Finite-volume Method - General Conservation Law

General Conservation Law

We can also consider the general conservation law problem, represented by the following PDE,

Here, represents a vector of states and represents the corresponding flux tensor. Again we can sub-divide the spatial domain into finite volumes or cells. For a particular cell, we take the volume integral over the total volume of the cell, which gives,

\quad (9) \qquad \qquad \int _{v_{i}} {{\partial {\mathbf u}} \over {\partial t}}\, dv + \int _{v_{i}} \nabla \cdot {\mathbf f}\left( {\mathbf u } \right)\, 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 {\bar u} }_{i} } \over {dt}} + \oint _{S_{i} } {\mathbf f} \left( {\mathbf u } \right) \cdot {\mathbf n }\ dS = {\mathbf 0},

where represents the total surface area of the cell and is a unit vector normal to the surface and pointing outward. So, finally, we are able to present the general result equivalent to (7), i.e.

\quad (11) \qquad \qquad {{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} .

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!

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