Jacobian-Free Incomplete Riemann Solvers

The purpose of this work is to present some recent developments about incomplete Riemann solvers for general hyperbolic systems. Polynomial Viscosity Matrix (PVM) methods based on internal approximations to the absolute value function are introduced, and they are compared with Chebyshev-based PVM solvers. 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 for which the Jacobians involve complex expressions. Some numerical experiments involving the relativistic magnetohydrodynamic equations are presented, both in one and two dimensions. The obtained results are in good agreement with those found in the literature and show that our schemes are robust and accurate, running stable under a satisfactory time step restriction.