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A direct Eulerian GRP scheme for relativistic hydrodynamics: Two-dimensional case

机译：相对论流体力学的直接欧拉GRP方案：二维情况

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摘要

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, P. He, H.Z. Tang, J. Comput. Phys. 230 (2011) 7964-7987] and the GRP scheme for the non-relativistic hydrodynamics [M. Ben-Artzi, J.Q. Li, G. Warnecke, J. Comput. Phys. 218 (2006) 19-43]. 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, J. Falcovitz, J. Comput. Phys. 55 (1984) 1-32] 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.
机译：本文针对二维（2D）相对论流体力学（RHD）开发了直接的欧拉广义黎曼问题（GRP）方案。它是针对一维（1D）RHD的GRP方案的扩展[Z.C.杨平Tang J.Comput。物理230（2011）7964-7987]和非相对论流体力学的GRP方案[M.本·阿齐（J.Q. Li，G。Warnecke，J.Comput。物理218（2006）19-43]。为了导出直接的欧拉GRP方案，必须使用相应的Riemann不变量和Rankine-Hugoniot跳跃条件直接求解欧拉公式中拆分的2D RHD方程的（局部）GRP，以便进行关键且精细的拉格朗日处理原始的GRP计划[M. Ben-Artzi，J。Falcovitz，J。Comput。物理55（1984）1-32]。从一维RHD方程或非相对论流体力学方程的GRP解析二维2D RHD方程的GRP的重要区别在于，在二维2D的GRP中穿过冲击波或稀疏波的流动区域RHD方程通过洛伦兹因子非线性耦合，该因子也是根据切向速度建立的。它是纯粹的多维相对论特征。给出了几个数值示例，以证明所提出的2D GRP方案的准确性和有效性。

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