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A Stability Analysis of Hybrid Schemes to Cure Shock Instability

Published online by Cambridge University Press:  03 June 2015

Zhijun Shen ,
Wei Yan  and
Guangwei Yuan
Zhijun Shen*
Affiliation:
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
Wei Yan*
Affiliation:
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
Guangwei Yuan*
Affiliation:
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
*
Corresponding author.Email:[email protected]
Email:[email protected]
Email:[email protected]
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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.

Keywords

35L6565M0876M1276L05Godunov methodsnumerical shock instabilityhybrid scheme
Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 5 , May 2014 , pp. 1320 - 1342
Copyright
Copyright © Global Science Press Limited 2014

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References

[1]Chandrashekar, P., Kinetic energy preserving and entropy stable finite volume schemes for compressible Euler and Navier-Stokes equations, Commun. Comput. Phys., 14 (2013), 1252–1286.Google Scholar
[2]Chauvat, Y., Moschetta, J. M. and Gressier, J., Shock wave numerical structure and the carbuncle phenomenon, Int. J. Numer. Meth. Fluids 47, 2005, 903–909.Google Scholar
[3]Davis, S.F., A rotationally biased upwind difference scheme for the Euler equations, J. Comput. Phys., 56 (1984), 65–92.Google Scholar
[4]Dumbser, M., Morschetta, J. M., Gressier, J., A matrix stability analysis of the carbuncle phenomenon, J. Comput. Phys., 197 (2004), 647–670.Google Scholar
[5]Dumbser, M. and Toro, E. F., On universal Osher-type schemes for general nonlinear hyperbolic conservation laws, Commun. Comput. Phys., 10(2011), 635–671.Google Scholar
[6]Godunov, S. K., A finite difference method for computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sb., 47 (1959), 271–306.Google Scholar
[7]Gressier, J. and Moschetta, J. M., Robustness versus accuracy in shock-wave computations, Int. J. Numer. Meth. Fluid., 33 (2000), 313–332.Google Scholar
[8]Harten, A. and Hyman, J. M., Self adjusting grid methods for one dimensional hyperbolic conservation laws, J. Comput. Phys., 50 (1983), 235–269.CrossRefGoogle Scholar
[9]Harten, A., Lax, P. D. and Leer, B. van, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Review, 25 (1983), 35–61.Google Scholar
[10]Janhunen, P., A positive conservative method for magnetohydrodynamics based on HLL and Roe methods, J. Comput. Phys., 166 (2000), 649–661.Google Scholar
[11]Kemm, F., A note on the carbuncle phenomenon in shallow water simulations, ZAMM – Journal of Applied Mathematics and Mechanics, DOI: 10.1002/zamm.201200176Google Scholar
[12]Kim, S. S., Kim, C., Rho, O. H. and Hong, S. K., Cures for the shock instability: development of a shock-stable Roe scheme, J. Comput. Phys., 185 (2003), 342–374.Google Scholar
[13]Kim, S. D., Lee, B. J., Lee, H. J., Jeung, I., Robust HLLC Riemann solver with weighted average flux scheme for strong shock, J. Comput. Phys., 228 (2009), 7634–7642.Google Scholar
[14]Kitamura, K., Roe, P. and Ismail, F., Evaluation of Euler Fluxes for Hypersonic Flow Computations, AIAA JOURNAL, 47(1) (2009), 44–53.CrossRefGoogle Scholar
[15]Levy, D. W., Powell, K. G., B. van Leer, Use of a rotated Riemann solver for the two-dimensional Euler equations, J. Comput. Phys., 106 (1993), 201–214.Google Scholar
[16]Q Li, J., Li, Q. B., Xu, K., Comparison of the generalized Riemann solver and the gas-kinetic scheme for inviscid compressible flow simulations, J. Comput. Phys., 230 (2011), 5080–5099.Google Scholar
[17]Liou, M. S. and Steffen, C. J., A new flux splitting scheme, J. Comput. Phys., 107 (1993), 23–39.Google Scholar
[18]Liou, M. S., Mass Flux schemes and connection to shock instability, J. Comput. Phys., 160 (2000), 623–648.Google Scholar
[19]Moschetta, J. M., Gressier, J., Robinet, J. C., Casalis, G., The Carbuncle Phenomenon: a Genuine Euler Instability?, in Godunov Methods, Theory and Applications, Ed. Toro, E.F., Kluwer Academic/Plenum Publ., (1995), 639–645.Google Scholar
[20]Nishikawa, H. and Kitamura, K., Very simple, carbuncle-Free, Boundary-Layer-Resolving, Rotated-Hybrid Riemann Solvers, J. Comput. Phys., 227 (2008), 2560–2581.Google Scholar
[21]Noh, W. F., Errors for calculations of strong shocks using an artificial viscosity and an artificial heat flux, J. Comput. Phys., 72 (1987) 78–120.CrossRefGoogle Scholar
[22]Pandolfi, M. and D’Ambrosio, D., Numerical instabilities in upwind methods: analysis and cures for the carbuncle phenomenon, J. Comput. Phys., 166 (2001), 271–301.Google Scholar
[23]Park, S. H., Kwon, J. H., On the dissipation mechanism of Godunov-type schemes, J. Comput. Phys., 188 (2003), 524–542.Google Scholar
[24]Peery, K. M. and Imlay, S. T., Blunt-Body Flow Simulations, AIAA Paper 88-2904, 1988.Google Scholar
[25]Quirk, J., A contribution to the Great Riemann Solver Debate, Int. J. Numer. Meth. Fluid., 18 (1994), 555–574.Google Scholar
[26]Ren, Y. X., A robust shock-capturing scheme based on rotated Riemann solvers, Computers & Fluids, 32 (2003), 1379–1403.Google Scholar
[27]Robinet, J., Gressier, J., Casalis, G. and Moschetta, J.-M., Shock wave instability and carbuncle phenomenon: same intrinsic origin?, J. Fluid Mech., 417 (2000), 237–263.CrossRefGoogle Scholar
[28]Roe, P. L., Approximate Riemann solvers, parameter vector and difference schemes, J. Comput. Phys., 43 (1981), 357–372.Google Scholar
[29]Sanders, R., Morano, E. and Druguet, M., Multidimensional dissipation for upwind schemes: stability and applications to gas dynamics, J. Comput. Phys., 145 (1998), 511–537.Google Scholar
[30]Scandaliato, A. L. and Liou, M. S.AUSM-based high-order solution for Euler equations. Commun. Comput. Phys., 12 (2012), 1096–1120.Google Scholar
[31]Toro, E. F., Spruce, M. and Speares, W., Restoration of the contact surface in the HLL-Riemann solver, Shock Wave, 4 (1994), 25–34.CrossRefGoogle Scholar
[32]Wada, Y. and Liou, M. S., An accurate and robust flux splitting scheme for shock and contact discontinuities, SIAM J. Sci. Comput., 18(3) (1997), 633–657.Google Scholar
[33]Wada, Y. and Liou, M. S., A Flux Splitting Scheme with High-Resolution and Robustness for Discontinuities, NASA T. M. 106452 (1994).Google Scholar
[34]Wu, H., Shen, L. J., Shen, Z. J., A hybrid numerical method to cure numerical shock instability, Commun. Comput. Phys., 8(5) (2010), 1264–1271.Google Scholar
[35]Xu, K., Li, Z., Dissipative mechanism in Godunov-type schemes, Int. J. Numer. Methods Fluids, 37 (2001), 1–22.Google Scholar