On Numerical Boundary Treatment of Hyperbolic Systems for Finite Difference and Finite Element Methods

It is well known that the stability of the initial-boundary value problem for a scalar equation does not necessarily imply stability for a vector equation with a similar boundary treatment. In fact, we show that for any boundary treatment one can construct systems for which the boundary treatment on the “natural” variables is not stable. Our main theorem is that one can introduce a treatment of the boundaries for which the stability of the system follows immediately from the stability for a scalar equation. This is accomplished by operating on the characteristic variables for those quantities that are not specified on the boundary. In a working code this can be accomplished by adding a correction term to the existing boundary algorithm. The analysis and computational results are presented for both finite differences and semi-discrete Galerkin methods.