Stability of Godunov’s method for a class of $2\times 2$ systems of conservation laws

Abstract:

We prove stability and convergence of the Godunov scheme for a special class of genuinely nonlinear $2 \times 2$ systems of conservation laws. The class of systems, which was identified and studied by Temple, is a subset of the class of systems for which the shock wave curves and rarefaction wave curves coincide. None of the equations of gas dynamics fall into this class, but equations of this type do arise, for example, in the study of multicomponent chromatography. To our knowledge this is the first time that a numerical method other than the random choice method of Glimm has been shown to be stable in the variation norm for a coupled system of nonlinear conservation laws. This implies that subsequences converge to weak solutions of the Cauchy problem, although convergence for $2 \times 2$ systems has been proved by DiPerna using the more abstract methods of compensated compactness.