一种鲁棒的HLLC格式及其稳定性分析

精确捕捉接触波和剪切波的Godunov型数值方法，如流行的HLLC格式，在模拟高超声速流动问题时会出现激波异常现象。对HLLC格式进行稳定性分析发现，流体主流方向的扰动都能有效衰减，但是横向的密度与剪切速度的扰动不会衰减。具有特殊对称性的二维Sedov爆轰波问题证明了横向通量和不稳定现象之间的密切联系。利用压力比和马赫数来探测数值激波层亚声速区的横向网格界面，并且在该界面的数值通量上增加熵波粘性和剪切波粘性来构造一种激波稳定的HLLC格式。分析表明，在熵波粘性和剪切波粘性的作用下，横向的所有扰动都会衰减。一系列数值测试证明了新格式不仅可以成功地抑制各类激波异常现象，还保留了原HLLC格式低耗散性的优点。

A robust HLLC scheme and stability analysis

Godunov-type numerical methods that can accurately capture contact wave and shear wave,such as the popular HLLC scheme,will cause shock anomalies when used in simulating hypersonic flow problems.The stability analysis of HLLC scheme shows that all perturbations in the flow direction are damped,but the perturbations of density and shear velocity in the transverse direction are undamped.The 2D Sedov blast wave problem with special symmetry demonstrates the close relationship between transverse flux and instability.The pressure ratio and Mach number are used to detect the transverse interfaces in the subsonic region of numerical shock layer,and a shock stable HLLC scheme is proposed by adding the viscosity of entropy wave and shear wave to numerical fluxes at these interfaces.The analysis shows that all perturbations in the transverse direction are damped with the help of viscosity of entropy wave and shear wave.A series of numerical tests show that the new scheme can not only successfully suppress all kinds of shock anomalies,but also retain the merit of low dissipation of the original HLLC scheme.