The Harten-Lax-van Leer with contact (HLLC) scheme is known to be plagued by various forms of numerical shock instabilities. In this paper, we propose a new framework for developing shock stable, contact and shear preserving approximate Riemann solvers based on the HLLC scheme for the Euler system of equations. The proposed framework termed as HLLC-SWM (Selective Wave Modified) scheme identifies and increases the magnitude of the inherent diffusive HLL component within the HLLC scheme in the vicinity of a shock wave while leaving its antidiffusive component unmodified to retain accuracy on linearly degenerate wavefields. We present two strategies to compute the requisite supplementary dissipation which results in HLLC-SWM-E and HLLC-SWM-P variants. Through a linear perturbation analysis of the HLLC-SWM framework, we clarify how the additional dissipation introduced in this way helps in damping of unphysical perturbations in primitive quantities under a derived CFL constraint. A matrix based stability analysis of a steady two-dimensional normal shock is used to show that both variants of the HLLC-SWM scheme are shock stable over a wide range of inlet Mach numbers. Results from standard test cases demonstrate that the HLLC-SWM schemes are capable of computing shock stable solutions on a variety of problems while retaining positivity and exact inviscid contact ability. On viscous flows, while the HLLC-SWM-P variant * Corresponding author is quite accurate, the HLLC-SWM-E variant introduces slight inaccuracy which can be corrected through a simple Mach number based switching function.
Various forms of numerical shock instabilities are known to plague many contact and shear preserving approximate Riemann solvers, including the popular Harten-Lax-van Leer with Contact (HLLC) scheme, during high speed flow simulations. In this paper we propose a simple and inexpensive novel strategy to prevent the HLLC scheme from developing such spurious solutions without compromising on its linear wave resolution ability. The cure is primarily based on a reinterpretation of the HLLC scheme as a combination of its well-known diffusive counterpart, the HLL scheme, and an antidiffusive term responsible for its accuracy on linear wavefields. In our study, a linear analysis of this alternate form indicates that shock instability in the HLLC scheme could be triggered due to the unwanted activation of the antidiffusive terms of its mass and interface-normal flux components on interfaces that are not aligned with the shock front. This inadvertent activation results in weakening of the favourable dissipation provided by its inherent HLL scheme and causes unphysical mass flux variations along the shock front. To mitigate this, we propose a modified HLLC scheme that employs a simple differentiable pressure based multidimensional shock sensor to achieve smooth control of these critical antidiffusive terms near shocks. Using a linear perturbation analysis and a matrix based stability analysis, we establish that the resulting scheme, called HLLC-ADC (Anti-Diffusion Control), is shock stable over a wide range of freestream Mach numbers. Results from standard numerical test cases demonstrate that the HLLC-ADC scheme is indeed free from the most common manifestations of shock instability including the Carbuncle phenomenon without significant loss of accuracy on shear dominated viscous flows.
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.
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