Commun. Comput. Phys., 15 (2014), pp. 1320-1342.


A Stability Analysis of Hybrid Schemes to Cure Shock Instability

Zhijun Shen 1*, Wei Yan 2, Guangwei Yuan 2

1 National Key Laboratory of Science and Technology on Computational Physics, Institute of Applied Physics and Computational Mathematics, P.O. Box 8009-26, Beijing 100088, China; Center for Applied Physics and Technology, HEDPS, Peking University, Beijing 100871, China.
2 National Key Laboratory of Science and Technology on Computational Physics, Institute of Applied Physics and Computational Mathematics, P.O. Box 8009-26, Beijing 100088, China.

Received 21 May 2013; Accepted (in revised version) 9 October 2013
Available online 14 February 2014
doi:10.4208/cicp.210513.091013a

Abstract

The carbuncle phenomenon has been regarded as a spurious solution produced by most of contact-preserving methods. The hybrid method of combining high resolution flux with more dissipative solver is an attractive attempt to cure this kind of non-physical phenomenon. In this paper, a matrix-based stability analysis for 2-D Euler equations is performed to explore the cause of instability of numerical schemes. By combining the Roe with HLL flux in different directions and different flux components, we give an interesting explanation to the linear numerical instability. Based on such analysis, some hybrid schemes are compared to illustrate different mechanisms in controlling shock instability. Numerical experiments are presented to verify our analysis results. The conclusion is that the scheme of restricting directly instability source is more stable than other hybrid schemes.

AMS subject classifications: 35L65, 65M08, 76M12, 76L05

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Key words: Godunov methods, numerical shock instability, hybrid scheme.

*Corresponding author.
Email: shen_zhijun@iapcm.ac.cn (Z. Shen), wyanmath01@sina.com (W. Yan), yuan_guangwei@iapcm.ac.cn (G. Yuan)
 

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