Approximate Osher-Solomon Schemes for Hyperbolic Systems

Abstract

The Osher-Solomon scheme is a classical Riemann solver which enjoys a number of interesting features: it is nonlinear, complete, robust, entropy-satisfying, smooth, etc. However, its practical implementation is rather cumbersome, computationally expensive, and applicable only to certain systems (compressible Euler equations for ideal gases or shallow water equations, for example). In this work, a new class of approximate Osher-Solomon schemes for the numerical approximation of general conservative and nonconservative hyperbolic systems is proposed. They are based on viscosity matrices obtained by polynomial or rational approximations to the Jacobian of the flux evaluated at some average states, and only require a bound on the maximal characteristic speeds. These methods are easy to implement and applicable to general hyperbolic systems, while at the same time they maintain the good properties of the original Osher-Solomon solver. The numerical tests indicate that the schemes are robust, running stable and accurate with a satisfactory time step restriction, and the computational cost is very advantageous with respect to schemes using a complete spectral decomposition of the Jacobians.