Editing Finite volume method (section)

== General conservation law ==

We can also consider the general [[Conservation law (physics)|conservation law]] problem, represented by the following [[partial differential equation|PDE]],

{{NumBlk|:|<math> \frac{\partial \mathbf u}{\partial t} + \nabla  \cdot {\mathbf f}\left( {\mathbf u } \right) = {\mathbf 0} . </math>|{{EquationRef|8}}}}

Here, <math> \mathbf u </math> represents a vector of states and <math>\mathbf f </math> represents the corresponding [[flux]] tensor. Again we can sub-divide the spatial domain into finite volumes or cells. For a particular cell, <math>i </math>, we take the volume integral over the total volume of the cell, <math>v _{i} </math>, which gives,

{{NumBlk|:|<math> \int _{v_{i}}  \frac{\partial \mathbf u}{\partial t}\, dv 
+ \int _{v_{i}}  \nabla  \cdot {\mathbf f}\left( {\mathbf u } \right)\, dv = {\mathbf 0} .</math>|{{EquationRef|9}}}}

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

{{NumBlk|:|<math>
v_{i} {{d {\mathbf {\bar u} }_{i} } \over dt} + \oint _{S_{i} } 
 {\mathbf f} \left( {\mathbf u } \right) \cdot {\mathbf n }\  dS  = {\mathbf 0}, </math>|{{EquationRef|10}}}}

where <math> S_{i} </math> represents the total surface area of the cell and <math>{\mathbf n}</math> is a unit vector normal to the surface and pointing outward. So, finally, we are able to present the general result equivalent to ({{EquationNote|8}}), i.e.

{{NumBlk|:|<math>
{{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} .</math>|{{EquationRef|11}}}}

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 scheme|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 always another cell's gain''!