# Some Background on Finite Volume Methods

We are generally interested in solving PDE's of the form

For the moment, let's focus our attention even further, on one of the simplest PDE's of that form, known as Burger's equation (inviscid form).

and let the numerical flux **F** be given by

Integrating the conservation form of Burger's eqn. and dividing by total volume gives,

If we can find a way to estimate the numerical fluxes and using information from the current time step, we will have a method for advancing the solution in time. Specifically

To see how this can be implemented in practice, go to the next section

## Roe Solver

One technique for estimating these numerical fluxes, developed by Roe, involves linearizing the system by evaluating the Jacobian matrix
at and approximating
. The original equation then becomes
. If the basis of eigenvectors of **A** is chosen, the system is uncoupled and is reduced to a series of 1D equations of the form

where is the eigenvalue corresponding to the eigenvector . Given initial data

the value at at subsequent times is if and if . The component of the numerical flux at is then given by

The two results can be combined in the following expression

Summing over all , the final result is