The purpose of this paper is twofold. First, the application of high-order methods in combination with the popular HLLC Riemann solver demonstrates that the grid-aligned shock instability can strongly affect simulation results when the grid resolution is increased. Beyond the well-documented two-dimensional behavior, the problem is particularly troublesome with three-dimensional simulations. Hence, there is a need for shock-stable modifications of HLLC-type solvers for high-speed flow simulations.

Second, the paper provides a stabilization of the popular HLLC flux based on a recently proposed mechanism for grid aligned-shock instabilities Fleischmann et al. (2020) [8]. The instability was found to be triggered by an inappropriate scaling of acoustic and advection dissipation for local low Mach numbers. These low Mach numbers occur during the calculation of fluxes in transverse direction of the shock propagation, where the local velocity component vanishes. A centralized formulation of the HLLC flux is provided for this purpose, which allows for a simple reduction of nonlinear signal speeds. In contrast to other shock-stable versions of the HLLC flux, the resulting HLLC-LM flux reduces the inherent numerical dissipation of the scheme.

The robustness of the proposed scheme is tested for a comprehensive range of cases involving strong shock waves. Three-dimensional single- and multi-component simulations are performed with high-order methods to demonstrate that the HLLC-LM flux also copes with latest challenges of compressible high-speed computational fluid dynamics.

本文的目的是双重的。首先，将高阶方法与流行的HLLC Riemann求解器结合使用证明，当提高网格分辨率时，网格对齐的冲击不稳定性会严重影响仿真结果。除了有据可查的二维行为外，对于三维仿真，问题尤其棘手。因此，需要对用于高速流动模拟的HLLC型求解器进行冲击稳定的修改。

其次，本文基于最近提出的网格对准电击不稳定性机制Fleischmann等，提供了流行的HLLC通量的稳定化方法。（2020）[8]。对于局部低马赫数，发现不稳定性是由声学和对流耗散的不适当缩放引起的。这些低马赫数发生在冲击传播横向方向上的通量计算过程中，局部速度分量消失了。为此，提供了HLLC通量的集中公式，可以简化非线性信号速度。与HLLC磁通的其他冲击稳定版本相比，所得HLLC-LM磁通减少了该方案的固有数值耗散。

在涉及强冲击波的各种情况下，对所提出方案的鲁棒性进行了测试。用高阶方法进行了三维单组分和多组分仿真，以证明HLLC-LM通量还可以应对可压缩高速计算流体动力学的最新挑战。