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Summary: The HLLEM scheme is a popular contact and shear preserving approximate Riemann solver that is known to be plagued by various forms of numerical shock instability. In this paper, we clarify that the shock instability exhibited by this scheme is primarily triggered by the spurious activation of the antidiffusive terms present in the first and third Riemann flux components on the transverse interfaces adjoining the shock front due to numerical perturbations. These erroneously activated terms are shown to counteract the favorable damping mechanism provided by its inherent HLL‐type diffusive terms, causing an unphysical variation of the conserved quantity ρu both along and across the numerical shock. To prevent this, two distinct strategies are proposed termed as Selective Wave Modification and Anti Diffusion Control. The former focuses on enhancing the quantity of the favorable HLL‐type dissipation available on these critical flux components by carefully increasing the magnitudes of certain nonlinear wave speed estimates, while the latter focuses on directly controlling the magnitude of these critical antidiffusive terms. A linear perturbation analysis is performed to gauge the effectiveness of these cures and to estimate a von Neumann–type stability bounds on the CFL number associated with their use. Results from a variety of classic shock instability test cases show that the proposed strategies are able to provide excellent shock stable solutions even on grids that are highly elongated across the shock front without compromising the accuracy on inviscid contact or shear dominated viscous flows. Numerical shock instability in the HLLEM scheme is found to be triggered by the erroneous activation of certain antidiffusive terms, within the numerical shock region, which counteract the stabilizing action of the inherent HLL‐type diffusive terms. Two distinct strategies named 'Selective Wave Modification' and 'Anti Diffusion Control' are proposed to avert the instability. The schemes resulting from these, are found to be robust on a wide variety of problems involving a wide range of grid aspect ratios. [ABSTRACT FROM AUTHOR]

Abstract The Harten–Lax–van Leer with contact (HLLC) approximate Riemann solver is known to be plagued by various forms of numerical shock instability. In this paper, we propose a new framework for developing shock stable versions of the HLLC scheme for the Euler system of equations without compromising on its contact and shear preserving ability. The proposed framework, termed as S elective W ave M odification (SWM), identifies and enhances the magnitude of the inherent HLL-type diffusive component within the HLLC scheme in the vicinity of a normal shock front. This is achieved by suitably increasing the magnitudes of the nonlinear wave speed estimates appearing in them through a dissipation parameter ϵ. A linear perturbation analysis is performed to gauge the effectiveness of the proposed framework in damping unwanted perturbations in physical quantities and is also used to estimate a von-Neumann-type stability bound on the CFL number associated with its use. Two distinct methods to estimate ϵ , which results in HLLC-SWM-E (Eigenvalue based) and HLLC-SWM-P (Pressure based) schemes, are proposed and compared. Further, a matrix based stability analysis is used to show that the proposed schemes can be configured to remain shock stable over a wide range of free stream Mach numbers. Numerical results confirm that the proposed schemes are capable of computing shock stable solutions for a wide variety of problems. On viscous flows, the HLLC-SWM-P variant is found to be the most accurate. Performance of these schemes suggests that it is possible to construct shock stable upwind schemes that simultaneously retains accuracy on the linearly degenerate wavefields. Highlights • A simple strategy to cure shock instability in the HLLC Riemann solver proposed. • Cure achieved by enhancing the inherent HLL-type diffusive component. • Two shock stable variants namely HLLC-SWM-E and HLLC-SWM-P are developed. • Robustness and accuracy of the scheme are demonstrated on several numerical examples. [ABSTRACT FROM AUTHOR]