An Improved Exact Riemann Solver for Relativistic Hydrodynamics

TL;DR

This paper introduces a new method for solving the relativistic Riemann problem exactly, eliminating initial guess ambiguities and improving computational efficiency in one-dimensional relativistic hydrodynamics simulations.

Contribution

The paper presents an analytic and numerical approach to determine wave patterns in relativistic Riemann problems without initial guesses, enhancing solver accuracy and efficiency.

Findings

Analytic solution for wave pattern determination.

Root of nonlinear equation can be found analytically for rarefaction waves.

Method improves the efficiency of relativistic hydrodynamics simulations.

Abstract

A Riemann problem with prescribed initial conditions will produce one of three possible wave patterns corresponding to the propagation of the different discontinuities that will be produced once the system is allowed to relax. In general, when solving the Riemann problem numerically, the determination of the specific wave pattern produced is obtained through some initial guess which can be successively discarded or improved. We here discuss a new procedure, suitable for implementation in an exact Riemann solver in one dimension, which removes the initial ambiguity in the wave pattern. In particular we focus our attention on the relativistic velocity jump between the two initial states and use this to determine, through some analytic conditions, the wave pattern produced by the decay of the initial discontinuity. The exact Riemann problem is then solved by means of calculating the root of a nonlinear equation. Interestingly, in the case of two rarefaction waves, this root can even be found analytically. Our procedure is straightforward to implement numerically and improves the efficiency of numerical codes based on exact Riemann solvers.