Stable and unstable numerical boundary conditions for Galerkin approximations to hyperbolic systems

Abstract

The Galerkin method for first order hyperbolic systems is considered. This method is seen to be unstable in an example because of improper treatment of the boundary conditions in the discrete problem. The Galerkin method is modified to be stable for any well-posed initial-boundary value problem in one dimension. It is shown that the previous estimates of the error in this modified method can be improved when the system admits an energy conserving norm. A system in two space dimensions in a region with a corner is considered. It is shown how the construction of stable methods for such problems can be reduced to a very simple linear programming problem.