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

doi:10.1007/s10483-025-3242-9

Applied Mathematics and Mechanics (English Edition) ›› 2025, Vol. 46 ›› Issue (4): 723-744.doi: https://doi.org/10.1007/s10483-025-3242-9

收稿日期:2024-11-17 修回日期:2025-02-20 发布日期:2025-04-07

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

Hongping GUO¹^,², Xun WANG³^,⁴, Zhijun SHEN¹^,⁵^,^†()

^1.National Key Laboratory of Computational Physics, Institute of Applied Physics andComputational Mathematics, Beijing 100088, China
^2.Faculty of Mathematics, Baotou Teachers’ College, Baotou 014030, Inner Mongolia Autonomous Region, China
^3.School of Mathematical Sciences, Peking University, Beijing 100871, China
^4.PKU-Changsha Institute for Computing and Digital Economy, Changsha 410205, China
^5.Center for Applied Physics and Technology, Peking University, Beijing 100871, China

Received:2024-11-17 Revised:2025-02-20 Published:2025-04-07
Contact: Zhijun SHEN E-mail:shen_zhijun@iapcm.ac.cn
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)

摘要/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

中图分类号:

O354

. [J]. Applied Mathematics and Mechanics (English Edition), 2025, 46(4): 723-744.

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 (English Edition), 2025, 46(4): 723-744.

图/表 7

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

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

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