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The modified Rusanov scheme for solving the ultra-relativistic Euler equations. (English) Zbl 07440254

Summary: We consider the ultra-relativistic Euler equations for an ideal gas, which are described in terms of the pressure \(p\) and the spatial part \(\mathbf{u} \in \mathbb{R}^3\) 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 (\(\alpha_{i + \frac{1}{2}}^n\)) 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.

MSC:

65-XX Numerical analysis
35L45 Initial value problems for first-order hyperbolic systems
35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
76Y05 Quantum hydrodynamics and relativistic hydrodynamics
76-XX Fluid mechanics
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