求解真实天体物理流体的Riemann问题
Solving the Riemann Problem for Realistic Astrophysical Fluids
摘要：
我们提出了新的方法来解决黎曼问题，精确和近似的一般状态方程（EoS），以促进现实的建模和理解的天体物理流动。如果EoS是单调的，则无论EoS的物理有效性如何，都可以保证新的精确广义EoS Riemann解的存在唯一性.我方确认：（1）在计算理想气体欧拉方程时，新的广义EoS Riemann解与原精确Riemann解匹配，（2）在计算理想气体EoS时，新的Harten-Lax-van Leer-Contact（HLLC）广义EoS Riemann解与原HLLC Riemann解匹配;（3）新的HLLC广义EoS Riemann解的解逼近新的精确解。我们用两种方法来求解EoS，一种是用双线性插值法来插值二维EoS表，另一种是在运行时解析计算热力学变量。插值方法更通用，因为它可以与其他单调和现实的EoS一起工作，而这里介绍的分析EoS求解器与相对理想化的EoS一起工作。数值结果表明，两种EoS求解器的精度相当。我们用HLLC广义EoS Riemann解算器研究了这两种方法的效率，发现在测试问题中，解析EoS解算器的速度更快。然而，我们指出，这两个EoS求解器的组合可能会成为有利的，在一些特定的问题。在整个研究中，我们假设局部热平衡。
英文摘要：
We present new methods to solve the Riemann problem both exactly and approximately for general equations of state (EoS) to facilitate realistic modeling and understanding of astrophysical flows. The existence and uniqueness of the new exact general EoS Riemann solution can be guaranteed if the EoS is monotone regardless of the physical validity of the EoS. We confirm that: (1) the solution of the new exact general EoS Riemann solver and the solution of the original exact Riemann solver match when calculating perfect gas Euler equations; (2) the solution of the new Harten-Lax-van Leer-Contact (HLLC) general EoS Riemann solver and the solution of the original HLLC Riemann solver match when working with perfect gas EoS; and (3) the solution of the new HLLC general EoS Riemann solver approaches the new exact solution. We solve the EoS with two methods, one is to interpolate 2D EoS tables by the bi-linear interpolation method, and the other is to analytically calculate thermodynamic variables at run-time. The interpolation method is more general as it can work with other monotone and realistic EoS while the analytic EoS solver introduced here works with a relatively idealized EoS. Numerical results confirm that the accuracy of the two EoS solvers is similar. We study the efficiency of these two methods with the HLLC general EoS Riemann solver and find that analytic EoS solver is faster in the test problems. However, we point out that a combination of the two EoS solvers may become favorable in some specific problems. Throughout this research, we assume local thermal equilibrium.
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