# 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).

Though this class of equations may appear relatively straightforward, it turns out that many such PDE's lead to the formation of discontinuities, or shocks, even from smooth initial data. In order to solve PDE's whose solutions contain shocks, we must turn to methods which do not assume the continuity of the solution, unlike the more intuitive finite difference methods. One way in which we can make sense of such solutions is to think of the average value of the solution over a given cell, rather than the value at specific grid points. Let the cell describe the spatial region between and and the temporal region between and . The average value of the solution over this cell is

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