Numerical stability analysis of shock-capturing methods for strong shocks II: High-order finite-volume schemes

The shock instability problem commonly arises in flow simulations involving strong shocks, particularly when employing high-order shock-capturing schemes, limiting their application in hypersonic flow simulations. In the current study, the shock instability problem of the fifth-order finite-volume WENO scheme is investigated. To this end, the matrix stability analysis method for the fifth-order scheme is established. Such an analysis method is capable of providing quantitative insights into the stability of shock-capturing and helping elucidate the mechanism causing shock instabilities. Results from the stability analysis and numerical experiments reveal that even dissipative solvers, which are generally believed to be capable of capturing strong shocks stably, also suffer from the shock instability problem when the spatial accuracy is increased to the fifth- order. Further investigation indicates that this is because the quadratic polynomial degenerated from the WENO reconstruction is not applicable for reconstructing variables on the faces near the numerical shock structure. Moreover, the shock instability problem of the fifth-order scheme is demonstrated to be a multidimensional coupled problem. To capture strong shocks stably, it is necessary to have sufficient dissipation on both transverse faces and normal faces in the vicinity of strong shocks. The spatial location of the perturbation leading to instability is also clarified by the proposed matrix analysis and numerical experiments, revealing that the instability arises from the numerical shock structure for the fifth-order scheme. Additionally, the local characteristic decomposition is demonstrated to be helpful to mitigate shock instabilities in high-order cases. Based on these conclusions and the core idea of the MR-WENO scheme, a shock-stable high-order scheme, the MR-WENO-SD scheme, is proposed. Equipped with the hybrid HLLC/HLL solver, this scheme is capable of capturing strong shocks stably while maintaining high-order accuracy. A series of numerical experiments demonstrate its stability and resolution capabilities.