This paper extends the second-order accurate BGK finite volume schemes for the ultra-relativistic flow simulations [5] to the 1D and 2D special relativistic hydrodynamics with the Synge equation of state. It is shown that such 2D schemes are very time-consuming due to the moment integrals (triple integrals) so that they are no longer practical. In view of this, the simplified BGK (sBGK) schemes are presented by removing some terms in the approximate nonequilibrium distribution at the cell interface for the BGK scheme without loss of accuracy. They are practical because the moment integrals of the approximate distribution can be reduced to the single integrals by some coordinate transformations. The relations between the left and right states of the shock wave, rarefaction wave, and contact discontinuity are also discussed, so that the exact solution of the 1D Riemann problem could be derived and used for the numerical comparisons. Several numerical experiments are conducted to demonstrate that the proposed gas-kinetic schemes are accurate and stable. A comparison of the sBGK schemes with the BGK scheme in one dimension shows that the former performs almost the same as the latter in terms of the accuracy and resolution, but is much more efficient.

本文将用于超相对论流动模拟的二阶精确BGK有限体积方案[5]扩展到具有Synge状态方程的一维和二维特殊相对论流体动力学。结果表明，由于力矩积分（三重积分），这种2D方案非常耗时，因此不再实用。鉴于此，通过在BGK方案的单元接口处删除近似非平衡分布中的某些项，从而提供了简化的BGK（sBGK）方案，而不会降低准确性。它们是实用的，因为可以通过一些坐标变换将近似分布的矩积分减小到单个积分。还讨论了冲击波，稀疏波和接触不连续性的左右状态之间的关系，因此可以导出一维黎曼问题的精确解并将其用于数值比较。进行了几个数值实验，以证明所提出的气体动力学方案是准确和稳定的。sBGK方案与BGK方案在一个维度上的比较表明，前者在准确性和分辨率方面的表现与后者几乎相同，但效率更高。