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
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