The Numerical Simulation of Relativistic Fluid Flow with Strong Shocks

Abstract

In this review we present and analyze the performance of a Go-dunov type method applied to relativistic fluid flow. Our model equations are the corresponding Euler equations for special relativistic hydrodynamics. By choosing an appropriate vector of unknowns, the equations of special relativistic fluid dynamics (RFD) can be written as a hyperbolic system of conservation laws. We give a complete description of the spectral decomposition of the Jacobian matrices associated to the fluxes in each spatial direction, (see (Donat et al., 1998), for details), which is the essential ingredient of the Godunov-type numerical method we propose in this paper. We also review a numerical flux formula that avoids/reduces numerical difficulties appearing in the ultrarelativistic regime, (i.e., high Lorentz factors). Using the spectral decompositions in a fundamental way, we construct high order versions of the basic first order scheme described by Donat and Marquina in ((Donat, Marquina, 1996)). We study, as a sample, a paricular shock tube test where special difficulties arise. We show two dimensional simulations where strong shocks are present, including a supersonic jet stream in a strongly ultra-relativistic scenario.