Applied Mathematics and Mechanics >
A low Mach number asymptotic analysis of dissipation-reducing methods for curing shock instability
Received date: 2024-11-17
Revised date: 2025-02-20
Online published: 2025-04-07
Supported by
Project supported by the National Natural Science Foundation of China (Nos. 12471367 and 12361076) and the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region (Nos. NJZY19186, NJZY22036, and NJZY23003)
Copyright
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
Hongping GUO, Xun WANG, Zhijun SHEN . A low Mach number asymptotic analysis of dissipation-reducing methods for curing shock instability[J]. Applied Mathematics and Mechanics, 2025 , 46(4) : 723 -744 . DOI: 10.1007/s10483-025-3242-9
| [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) |
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