New Types of Jacobian-Free Approximate Riemann Solvers for Hyperbolic Systems

Abstract

We present recent advances in PVM (Polynomial Viscosity Matrix) methods based on internal approximations to the absolute value function. These solvers only require a bound on the maximum wave speed, so no spectral decomposition is needed. Moreover, they can be written in Jacobian-free form, in which only evaluations of the physical flux are used. This is particularly interesting when considering systems with complex Jacobians, as the relativistic magnetohydrodynamics (RMHD) equations. The proposed solvers have also been extended to the case of approximate DOT (Dumbser-Osher-Toro) methods, which can be regarded as simple and efficient approximations to the classical Osher-Solomon method. Some numerical experiments involving the RMHD equations are presented. The obtained results are in good agreement with those found in the literature and show that our schemes are robust and accurate. Finally, notice that although this work focuses on RMHD, the proposed schemes can be directly applied to general hyperbolic systems.