一种简单的精确捕捉接触间断的黎曼求解器

基于Godunov型数值格式的有限体积法是求解双曲型守恒律系统的主流方法,其中用来计算界面数值通量的黎曼求解器在很大程度上决定了数值格式在计算中的表现。单波的Rusanov求解器和双波的HLL求解器具有简单、高效和鲁棒性好等优点,但是在捕捉接触间断时耗散太大。全波的HLLC格式能够精确捕捉接触间断,但是在计算中出现的激波不稳定现象限制了其在高马赫数流动问题中的应用。本文利用双曲正切函数和五阶WENO格式来重构界面两侧的密度值,并且结合边界变差下降算法来减小Rusanov格式耗散项中的密度差,从而提高格式对于接触间断的分辨率。研究表明,相比于全波的HLLC求解器,本文构造的黎曼求解器不仅具有更高的接触分辨率,而且还具有更好的激波稳定性。

A simple Riemann solver accurate for contact discontinuity

The finite volume method based on Godunov-type numerical scheme is the mainstream method for solving hyperbolic conservation law systems and the performance of the numerical scheme is largely determined by the Riemann solver for calculating the numerical flux at the cell interface.The one-wave Rusanov solver and the two-wave HLL solver have the advantages of simplicity,high efficiency and good robustness,but they are too dissipative in resolving the contact discontinuity.The complete-wave HLLC scheme can capture contact discontinuities accurately,but its applications in high Mach number flow problems are limited by the shock instability phenomenon. In this paper,the hyperbolic tangent function and the fifth-order WENO scheme are used to reconstruct the density values on both sides of the cell interface and the boundary variation diminishing algorithm is used to reduce the density difference in the dissipative term of Rusanov scheme,so as to improve the resolution for contact discontinuity significantly.Results of a series of numerical experiments demonstrate that compared with the complete-wave HLLC solver,the proposed Riemann solver has not only higher resolution but also better shock stability.