Positive schemes for solving multi-dimensional hyperbolic systems of conservation laws II

Abstract

Kurganov and Tadmor have developed a numerical scheme for solving the initial value problem for hyperbolic systems of conservation laws. They showed that in the scalar case their scheme satisfies a local maximum–minimum principle i.e., the solution at future is bounded above and below by the solution at current locally. In this paper we show that this scheme is positive in the sense of Friedrichs for systems as well. We present the scheme of Kurganov and Tadmor as a convex combination of composites of positive schemes. Since each component of a composite scheme is bounded in the l 2 norm, so is the convex combination of the composites. To achieve second order accuracy in time, we use a Runge–Kutta type scheme due to Shu and Osher. We present two numerical experiments to add to the ones carried out by Kurganov and Tadmor.