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A simple HLLE-type scheme is proposed for all Mach number flows. In the proposed scheme, no extra wave structure is added in the HLLE scheme to resolve the shear wave while the contact wave is resolved by adding a wave structure similar to the HLLEM scheme. The resolution of the shear layers and the flow features at low Mach numbers is achieved by a velocity reconstruction method based on the face normal Mach number. Robustness against the numerical instabilities is achieved by scaling the velocity reconstruction function in the vicinity of shock with a multi-dimensional pressure sensor. The ability of the proposed scheme to resolve low Mach flow features is demonstrated through asymptotic analysis while the stability of the proposed scheme for strong shock is demonstrated through linear perturbation and matrix stability analyses. A set of numerical test cases are solved to show that the proposed scheme is free from numerical shock instability problems at high speeds and is capable of resolving the flow features at very low Mach numbers. • A new velocity reconstruction method for an all Mach number HLLE scheme. • Velocity reconstruction based on face normal Mach number and pressure function. • Velocity reconstruction resolves shear layers and low Mach flow features while presrving shock stability. • HLLEM type method for resolving contact wave while preserving shock stability. • Analytical and numerical demonstration of all Mach capability of the proposed scheme. [ABSTRACT FROM AUTHOR]

The numerical shock instability problem significantly limits the application of the finite volume method in hypersonic flow simulations, especially for high-order schemes, which will encounter more severe shock instability problems. In the previous work, the MSAT (matrix stability analysis tool) was introduced to quantitatively analyze the numerical shock instability problem. However, the utilization of the three-point reconstruction stencil in the original version of MSAT presents challenges in conducting investigations into high-order schemes. The current work provides an extension of MSAT by extending the reconstruction stencil to five points, a challenging but necessary endeavor for high-order schemes. This extension enables MSAT to provide analyses for schemes with high-order reconstruction methods. Furthermore, the updated MSAT can be utilized to investigate the impact of characteristic variable reconstruction on the shock instability problem. Results demonstrate the reliability of the updated MSAT and reveal novel characteristics in the shock instability of high-order schemes. Consequently, the updated MSAT will contribute to the investigation of shock instabilities for high-order schemes, ultimately aiding in mitigating this challenging problem.

The simulation of supersonic or hypersonic flows often suffers from numerical shock instabilities if the flow field contains strong shocks, limiting the further application of shock-capturing schemes. In this paper, we develop the unified matrix stability analysis method for schemes with three-point stencils and present MSAT (matrix stability analysis tool), an open-source tool to quantitatively analyze the shock instability problem. Based on the finite-volume approach on the structured grid, MSAT can be employed to investigate the mechanism of the shock instability problem, evaluate the robustness of numerical schemes, and then help to develop robust schemes. Also, MSAT has the ability to analyze the practical simulation of supersonic or hypersonic flows, evaluate whether it will suffer from shock instabilities, and then assist in selecting appropriate numerical schemes accordingly. As a result, MSAT is a helpful tool that can investigate the shock instability problem and help to cure it.