Stable Galerkin Methods for Hyperbolic Systems

This paper considers the stability of the Galerkin method for first order hyperbolic systems. Just as in finite difference theory, many Galerkin methods, stable and convergent for the Cauchy problem, are unstable for hyperbolic boundary value problems because of improper treatment of the boundary conditions. The usual Galerkin method is seen to be unstable for a linear, constant coefficient, well-posed hyperbolic system on $[0,1]$ A method of treating boundary conditions is then proposed which will yield a stable and convergent method for any well-posed, linear, hyperbolic system in one dimension. This idea generalizes in a natural way to problems in more than one space dimension. Examples in one and two dimensions are considered. This paper extends the work of Gunzburger [Math. Comp., 31 (1977), pp. 671–675] to the general hyperbolic system.