The modified Rusanov scheme for solving the ultra-relativistic Euler equations

• Published in: European journal of mechanics, B, Fluids B, Fluids, 2021-11, Vol.90, p.89-98
• Main Authors: Mohamed, Kamel, Abdelrahman, Mahmoud A.E.
• Format: Article
• Language: English
• Subjects: Entropy shocks; Finite volume scheme; Modified Rusanov scheme; Rarefaction waves; Relativistic Euler equations; Riemann invariants; Riemann solutions
• ISSN: 0997-7546; 1873-7390
• DOI: 10.1016/j.euromechflu.2021.07.014

We consider the ultra-relativistic Euler equations for an ideal gas, which are described in terms of the pressure p and the spatial part u∈R3 of the dimensionless four-velocity. We also numerically investigate the ultra-relativistic Euler equations using the modified Rusanov scheme compared with classical Rusanov scheme. Our scheme consists of predictor and corrector stage. The predictor stage contains a parameter of control (αi+12n) is responsible for the numerical diffusion of this scheme. In order to control this parameter, we use a strategy depending on limiter theory and using Riemann invariants. The corrector stage recovers the balance conservation equation. The scheme can compute the numerical flux corresponding to the real state of solution without relying on Riemann problem solvers. The numerical results show the high resolution of the proposed finite volume scheme (modified Rusanov) and confirm its capability to provide accurate simulations for the ultra-relativistic Euler equations under regimes with strong shocks and rarefactions.