Using a Viscosity Matrix to Construct a Riemann Solver for the Equations of Special Relativistic Hydrodynamics

Abstract

Traditionally, to solve the hydrodynamic equations a Godunov method is used whose main stage is the solution of a Riemann problem to compute the fluxes of conservative variables through the interfaces of adjacent cells. Most numerical Riemann solvers are based on partial or full spectral decompositions of the Jacobian matrix with respect to the spatial derivatives. However, when using complex hyperbolic models and various types of equations of state, even partial spectral decompositions are quite difficult to find analytically. Such hyperbolic systems include the equations of special relativistic magnetohydrodynamics. In this paper, a numerical Riemann solver is constructed by means of a viscosity matrix based on Chebyshev polynomials. This scheme does not require any information about the spectral decomposition of the Jacobian matrix, while considering all types of waves in its design. To reduce the dissipation of the numerical solution, a piecewise parabolic reconstruction of the physical variables is used. The behavior of the numerical method is studied by using some classical test problems.