The modified Rusanov scheme for solving the ultra-relativistic Euler equations. (English) Zbl 1497.65146

Authors’ abstract: This paper considers the ultra-relativistic Euler equations for an ideal gas, which are described in terms of the pressure \(p\) and the spatial part \(u \in \mathbb{R}^3\) of the dimensionless four-velocity. First, the authors provide an exact solution to the Riemann problem for this system. Then, they turn to the main results of the paper, namely the derivation of a modified Rusanov scheme. The authors’ scheme consists of predictor and corrector stages. The predictor stage contains a parameter \(\alpha_{i + \frac 1 2}^n\), which is responsbible for the numerical diffusion. In order to control this parameter, the authors introduce a strategy depending on limiter theory and using Riemann invariants. The corrector stage projects back to the conservation equation. The scheme thus computes an approxmation of the Riemann problem solution. 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, especially compared to the traditional Rusanov scheme.


65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L45 Initial value problems for first-order hyperbolic systems
35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations
76Y05 Quantum hydrodynamics and relativistic hydrodynamics
76L05 Shock waves and blast waves in fluid mechanics
76N15 Gas dynamics (general theory)
35Q31 Euler equations
Full Text: DOI


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