A cure for numerical shock instability in HLLC Riemann solver using antidiffusion control

Abstract

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.

Strategies to cure numerical shock instability in the HLLEM Riemann solver

“…This mechanism of instability was found to closely resemble the one proposed for the HLLC scheme in, 20 probably indicating the generality of the mechanism for any HLL-based contact and shear preserving approximate Riemann solver. A linear scale analysis technique revealed that, on the critical transverse interfaces adjoining the normal shock front, both the inherent HLL-type diffusive terms and the antidiffusive terms present in the first and third Riemann flux components achieved by the HLLEM scheme are inadvertently activated due to these perturbations.…”

“…20 Note that this pressure-ratio based L,R also retains the merits of the shock sensor discussed in Section 6.1.1, ie, a multidimensional stencil and local solution dependence. A symmetrical definition holds for L,R i+1∕2, .…”

“…Suppose we employ the HLLE scheme to evaluate the transverse directional mass flux v and the transverse directional momentum flux uv in Equations (19) and (20). In the following, we analyze the behavior of the numerical dissipation made available through the HLLE discretization of each of these critical flux components separately.…”

“…20 They argued that, due to random numerical perturbations, these antidiffusive terms are inadvertently activated on these interfaces and causes a reduction in the stabilizing effect of the inherently present HLL-type diffusive terms that seeks to damp these perturbations. 20 They argued that, due to random numerical perturbations, these antidiffusive terms are inadvertently activated on these interfaces and causes a reduction in the stabilizing effect of the inherently present HLL-type diffusive terms that seeks to damp these perturbations.…”

An accurate and robust HLLC‐type Riemann solver for the compressible Euler system at various Mach numbers

“…This analysis approach is first proposed by Dumbser et al 45 and followed by many scholars 18,[46][47][48] to study the occurrence of unstable modes during the shock wave computation. This analysis approach is first proposed by Dumbser et al 45 and followed by many scholars 18,[46][47][48] to study the occurrence of unstable modes during the shock wave computation.…”