A Modified Roe Scheme and Stability Analysis

HU Lijun; ZHAO Kunlei

doi:10.21656/1000-0887.400388

Volume 41 Issue 10

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HU Lijun, ZHAO Kunlei. A Modified Roe Scheme and Stability Analysis[J]. Applied Mathematics and Mechanics, 2020, 41(10): 1110-1124. doi: 10.21656/1000-0887.400388

Citation:

HU Lijun, ZHAO Kunlei. A Modified Roe Scheme and Stability Analysis[J]. Applied Mathematics and Mechanics, 2020, 41(10): 1110-1124. doi: 10.21656/1000-0887.400388

Citation:

HU Lijun, ZHAO Kunlei. A Modified Roe Scheme and Stability Analysis[J]. Applied Mathematics and Mechanics, 2020, 41(10): 1110-1124. doi: 10.21656/1000-0887.400388

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A Modified Roe Scheme and Stability Analysis

doi: 10.21656/1000-0887.400388

HU Lijun¹,
ZHAO Kunlei²

¹College of Mathematics and Statistics, Hengyang Normal University,Hengyang, Hunan 421002, P.R.China;²LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences,Beijing 100190, P.R.China

Received Date: 2019-12-30
Rev Recd Date: 2020-02-08
Publish Date: 2020-10-01

Abstract

Abstract

Low dissipation shock-capturing methods, including the popular Roe scheme, will encounter the shock instability phenomenon in the computation of the multidimensional strong shock wave problems. This will seriously affect the schemes’ accurate simulation of the hypersonic flow problems. The small perturbation analysis of the Roe scheme was carried out. The results show that, all perturbations in the longitudinal direction of the shock front are damped, but the perturbations of density and shear velocity in the transverse direction are undamped. The viscosities corresponding to the entropy wave and shear wave were added to the flux transverse to the shock front to suppress instability of the Roe scheme. To prevent the improper viscosity from influencing the resolution of contact discontinuity and shear layers, 2 switching functions were defined, so that the viscosity was only added to the transverse flux in the subsonic region of the shock layer. The numerical tests show that, the modified Roe scheme not only retains the merit of high resolution of the original Roe scheme, but also has better robustness and eliminates the shock wave instability.
- hypersonic flow,
- Roe scheme,
- small perturbation analysis,
- numerical viscosity,
- shock instability

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References(24)

References

[1]	ROE P L. Approximate Riemann solvers, parameter vectors, and difference schemes[J]. Journal of Computational Physics,1981,43(2): 357-372.
[2]	TORO E F, SPRUCE M, SPEARES W. Restoration of the contact surface in the HLL Riemann solver[J]. Shock Waves,1994,4(1): 25-34.
[3]	QUIRK J J. A contribution to the great Riemann solver debate[J]. International Journal for Numerical Methods in Fluids,1994,18(6): 555-574.
[4]	GRESSIER J, MOSCHETTA J M. Robustness versus accuracy in shock-wave computations[J]. International Journal for Numerical Methods in Fluids,2000,33(3): 313-332.
[5]	SHEN Z J, YAN W, YUAN G W. A robust HLLC-type Riemann solver for strong shock[J]. Journal of Computational Physics,2016,309: 185-206.
[6]	XIE W J, LI H, TIAN Z Y, et al. A low diffusion flux splitting method for inviscid compressible flows[J]. Computers & Fluids,2015,112(2): 83-93.
[7]	LIOU M S. Mass flux scheme and connection to shock instability[J]. Journal of Computational Physics,2000,160(2): 623-648.
[8]	LIOU M S. A sequel to AUSM: AUSM+[J]. Journal of Computational Physics,1996,129(2): 364-382.
[9]	XU K, LI Z W. Dissipative mechanism in Godunov-type schemes[J]. International Journal of Numerical Methods in Fluids,2001,37(1): 1-22.
[10]	DUMBSER M, MOSCHETTA J M, GRESSIER J. A matrix stability analysis of the carbuncle phenomenon[J]. Journal of Computational Physics,2004,197(2): 647-670.
[11]	CHEN S S, YAN C, LIN B X, et al. Affordable shock-stable item for Godunov-type schemes against carbuncle phenomenon[J]. Journal of Computational Physics,2018,373: 662-672.
[12]	XIE W J, LI W, LI H, et al. On numerical instabilities of Godunov-type schemes for strong shocks[J]. Journal of Computational Physics,2017,350: 607-637.
[13]	SIMON S, MANDAL J C. A cure for numerical shock instability in HLLC Riemann solver using antidiffusion control[J]. Computers & Fluids,2018,174: 144-166.
[14]	WU H, SHEN L J, SHEN Z J. A hybrid numerical method to cure numerical shock instability[J]. Communications in Computational Physics,2010,8: 1264-1271.
[15]	REN Y X. A robust shock-capturing scheme based on rotated Riemann solvers[J]. Computers & Fluids,2003,32(10): 1379-403.
[16]	NISHIKAWA H, KITAMURA K. Very simple, carbuncle-free, boundary-layer-resolving, rotated-hybrid Riemann solvers[J]. Journal of Computational Physics,2008,227(4): 2560-2581.
[17]	PEERY K M, IMLAY S T. Blunt-body flow simulations[C]//24th Joint Propulsion Conference . Boston, MA, 1988.
[18]	FLEISCHMANN N, ADAMI S, HU X Y, et al. A low dissipation method to cure the grid-aligned shock instability[J]. Journal of Computational Physics,2020,401: 109004.
[19]	KIM S, KIM C, RHO O H, et al. Cures for the shock instability: development of a shock-stable Roe scheme[J]. Journal of Computational Physics,2003,185(2): 342-374.
[20]	RODIONOV A. Artificial viscosity in Godunov-type schemes to cure the carbuncle phenomenon[J]. Journal of Computational Physics,2017,345: 308-329.
[21]	RODIONOV A. Artificial viscosity to cure the shock instability in high-order Godunov-type schemes[J]. 〖JP〗 Computers & Fluids,2019,190: 77-97.
[22]	RODIONOV A. Artificial viscosity to cure the carbuncle phenomenon: the three-dimensional case[J]. Journal of Computational Physics,2018,361: 50-55.
[23]	WOODWARD P, COLELLA P. The numerical simulation of two-dimensional fluid flow with strong shocks[J]. Journal of Computational Physics,1984,54(1): 115-173.
[24]	HUANG K B, WU H, YU H, et al. Cures for numerical shock instability in HLLC solver[J]. International Journal of Numerical Methods in Fluids,2011,65(9): 1026-1038.