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A direct Eulerian GRP scheme for relativistic hydrodynamics: two-dimensional case. (English) Zbl 1408.76598
Summary: This paper develops a direct Eulerian generalized Riemann problem (GRP) scheme for two-dimensional (2D) relativistic hydrodynamics (RHD). It is an extension of the GRP scheme for one-dimensional (1D) RHDs [Z.-C. Yang et al., J. Comput. Phys. 230, No. 22, 7964–7987 (2011; Zbl 1408.76597)] and the GRP scheme for the non-relativistic hydrodynamics [M. Ben-Artzi et al., J. Comput. Phys. 218, No. 1, 19–43 (2006; Zbl 1158.76375)]. In order to derive the direct Eulerian GRP scheme, the (local) GRP of the split 2D RHD equations in the Eulerian formulation has to be directly resolved by using corresponding Riemann invariants and Rankine-Hugoniot jump conditions so that the crucial and delicate Lagrangian treatment in the original GRP scheme [M. Ben-Artzi und J. Falcovitz, J. Comput. Phys. 55, 1–32 (1984; Zbl 0535.76070)] may be avoided. An important difference of resolving the GRP of the split 2D RHD equations from the GRP of the 1D RHD equations or the non-relativistic hydrodynamical equations is coming from the fact that the flow regions across the shock or rarefaction wave in the GRP of the split 2D RHD equations are nonlinearly coupled through the Lorentz factor which is also built in terms of the tangential velocities. It is a purely multi-dimensional relativistic feature. Several numerical examples are given to demonstrate the accuracy and effectiveness of the proposed 2D GRP scheme.

##### MSC:
 76Y05 Quantum hydrodynamics and relativistic hydrodynamics
RAM; WHAM
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