ABSTRACT
This chapter considers the numerical approximation for the 1-D ultra-relativistic Euler (URE) equations. These equations describe an ideal gas in terms of the particle density n, the spatial part of the four-velocity u, and the pressure p. Due to the inherently nonlinear and complex nature emerging in relativistic flows, the classical Euler equations are not capable of stimulating fluid flows near the speed of light. The high-resolution shock capturing cone grid and wave-front tracking schemes are implemented in order to solve the URE equations.
We also consider the Riemann invariants and simulate them. The robustness and efficiency of the proposed schemes are exhibited by the numerical results. We also calculate the experimental order of convergence and numerical $L^1$-stability of the schemes.
