A low Mach number asymptotic analysis of dissipation-reducing methods for curing shock instability

doi:10.1007/s10483-025-3242-9

Abstract

Abstract:

We are intrigued by the issues of shock instability, with a particular emphasis on numerical schemes that address the carbuncle phenomenon by reducing dissipation rather than increasing it. For a specific class of planar flow fields where the transverse direction exhibits vanishing but non-zero velocity components, such as a disturbed one-dimensional (1D) steady shock wave, we conduct a formal asymptotic analysis for the Euler system and associated numerical methods. This analysis aims to illustrate the discrepancies among various low-dissipative numerical algorithms. Furthermore, a numerical stability analysis of steady shock is undertaken to identify the key factors underlying shock-stable algorithms. To verify the stability mechanism, a consistent, low-dissipation, and shock-stable HLLC-type Riemann solver is presented.

Key words: Riemann solver, numerical shock instability, low Mach number, HLLC

2010 MSC Number:

O354

Hongping GUO, Xun WANG, Zhijun SHEN. A low Mach number asymptotic analysis of dissipation-reducing methods for curing shock instability. Applied Mathematics and Mechanics (English Edition), 2025, 46(4): 723-744.

TrendMD

Figures/Tables 7

Fig. 1

Fig. 2

Table 1

Maximum real parts of the eigenvalues for the LM-ex scheme with different ϕk"

$(ϕ 1, ϕ 2, ϕ 3, ϕ 4)$	$max (Re (λ))$	$(ϕ 1, ϕ 2, ϕ 3, ϕ 4)$	$max (Re (λ))$
$(1, ϕ 0, 1, 1)$	$− 0.078 040$	$(1, 1, 1, 1)$	0.741 420
$(1, ϕ 0, 1, ϕ 0)$	$− 0.064 242$	$(1, 1, ϕ 0, 1)$	0.741 504
$(ϕ 0, ϕ 0, 1, 1)$	$− 0.061 772$	$(1, 1, ϕ 0, ϕ 0)$	$> 2$
$(1, ϕ 0, ϕ 0, 1)$	$− 0.031 090$	$(ϕ 0, 1, ϕ 0, 1)$	$> 2$
$(ϕ 0, ϕ 0, ϕ 0, 1)$	$− 0.028 918$	$(ϕ 0, 1, 1, ϕ 0)$	$> 2$
$(1, ϕ 0, ϕ 0, ϕ 0)$	$− 0.019 960$	$(ϕ 0, 1, ϕ 0, ϕ 0)$	$> 2$
$(ϕ 0, ϕ 0, 1, ϕ 0)$	$− 0.007 735$	$(1, 1, 1, ϕ 0)$	$> 2$
$(ϕ 0, ϕ 0, ϕ 0, ϕ 0)$	$− 0.000 002$	$(ϕ 0, 1, 1, 1)$	$> 2$

Table 1

Fig. 3

Fig. 4

Fig. 5

Fig. 6

References 30

[1]	ROE, P. L. Approximate Riemann solvers, parameter vectors, and difference schemes. Journal of Computational Physics, 43, 357–372 (1981)
[2]	TORO, E. F., SPRUCE, M., and SPEARES, W. Restoration of the contact surface in the HLL-Riemann solver. Shock Waves, 4(1), 25–34 (1994)
[3]	GODUNOV, S. K. A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics. Matematicheskii Sbornik, 47, 357–393 (1959)
[4]	QUIRK, J. J. A contribution to the great Riemann solver debate. International Journal for Numerical Methods in Fluids, 18(6), 555–574 (1994)
[5]	LIOU, M. S. Mass flux schemes and connection to shock instability. Journal of Computational Physics, 160, 623–648 (2000)
[6]	XU, K. and LI, Z. W. Dissipative mechanism in Godunov-type schemes. International Journal for Numerical Methods in Fluids, 37, 1–22 (2001)
[7]	HARTEN, A., LAX, P. D., and LEER, B. V. On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws, Springer Berlin Heidelberg, Berlin, 53–79 (1997)
[8]	SIMON, S. and MANDAL, J. C. A simple cure for numerical shock instability in the HLLC Riemann solver. Journal of Computational Physics, 378, 477–496 (2019)
[9]	PANDOLFI, M. and D’AMBROSIO, D. Numerical instabilities in upwind methods: analysis and cures for the carbuncle phenomenon. Journal of Computational Physics, 166, 271–301 (2001)
[10]	SANDERS, R., MORANO, E., and DRUGUET, M. C. Multidimensional dissipation for upwind schemes: stability and applications to gas dynamics. Journal of Computational Physics, 145, 511–537 (1998)
[11]	SHEN, Z. J., YAN, W., and YUAN, G. W. A robust HLLC-type Riemann solver for strong shock. Journal of Computational Physics, 309, 185–206 (2016)
[12]	HU, L. J., YUAN, H. Z., and ZHAO, K. L. A shock-stable HLLEM scheme with improved contact resolving capability for compressible Euler flows. Journal of Computational Physics, 453, 110947 (2022)
[13]	LIU, L. Q., LI, X., and SHEN, Z. J. Overcoming shock instability of the HLLE-type Riemann solvers. Journal of Computational Physics, 418, 109628 (2020)
[14]	KEMM, F. Heuristical and numerical considerations for the carbuncle phenomenon. Applied Mathematics and Computation, 320, 596–613 (2018)
[15]	CHEN, Z. Q., HUANG, X. D., REN, Y. X., XIE, Z. F., and ZHOU, M. Mechanism study of shock instability in Riemann-solver-based shock-capturing scheme. AIAA Journal, 56(9), 3636–3651 (2018)
[16]	CHEN, Z. Q., HUANG, X. D., REN, Y. X., XIE, Z. F., and ZHOU, M. Mechanism-derived shock instability elimination for Riemann-solver-based shock-capturing scheme. AIAA Journal, 56(9), 3652–3666 (2018)
[17]	XIE, W. J., TIAN, Z. Y., ZHANG, Y., HANG, Y., and REN, W. J. Further studies on numerical instabilities of Godunov-type schemes for strong shocks. Computers & Mathematics with Applications, 102, 65–86 (2021)
[18]	FLEISCHMANN, N., ADAMI, S., HU, X. Y., and ADAMS, N. A. A low dissipation method to cure the grid-aligned shock instability. Journal of Computational Physics, 401, 109004 (2020)
[19]	FLEISCHMANN, N., ADAMI, S., and ADAMS, N. A. A shock-stable modification of the HLLC Riemann solver with reduced numerical dissipation. Journal of Computational Physics, 423, 109762 (2020)
[20]	GUILLARD, H. and VIOZAT, C. On the behaviour of upwind schemes in the low Mach number limit. Computers & Fluids, 28, 63–86 (1999)
[21]	GUILLARD, H. and MURRONE, A. On the behavior of upwind schemes in the low Mach number limit: II. Godunov type schemes. Computers & Fluids, 33, 655–675 (2004)
[22]	GUILLARD, H. and NKONGA, B. On the behaviour of upwind schemes in the low Mach number limit: a review. Handbook of Numerical Analysis, 18, 203–231 (2017)
[23]	DUMBSER, M., MORSCHETTA, J. M., and GRESSIER, J. A matrix stability analysis of the carbuncle phenomenon. Journal of Computational Physics, 197, 647–670 (2004)
[24]	EINFELDT, B., MUNZ, C. D., ROE, P. L., and SJOGREEN, B. On Godunov-type methods near low density. Journal of Computational Physics, 92, 273–295 (1991)
[25]	RIEPER, F. On the Behaviour of Numerical Schemes in the Low Mach Number Regime, Ph.D. dissertation, Brandenburg Technical University, Cottbus (2008)
[26]	KLEIN, R. Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics, I, one-dimensional flow. Journal of Computational Physics, 121, 213–217 (1995)
[27]	LANDAU, L. D. and LIFSHITZ, E. M. Fluid Mechanics, 2nd. ed., Pergamon Press, Oxford (1987)
[28]	LUBCHICH, A. A. and PUDOVKIN, M. I. Interaction of small perturbations with shock waves. Physics of Fluids, 16, 4489–4505 (2004)
[29]	MCKENZIE, J. F. and WESTPHAL, K. O. Interaction of linear waves with oblique shock waves. Physics of Fluids, 11, 2350–2362 (1968)
[30]	SHEN, Z. J., YAN, W., and YUAN, G. W. A stability analysis of hybrid schemes to cure shock instability. Communications in Computational Physics, 15, 1320–1342 (2014)