A Hybrid Scheme of Rotational Flux for Solving 2D Euler Equations

JIA Dou; ZHENG Supei

doi:10.21656/1000-0887.410196

Volume 42 Issue 2

Feb. 2021

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JIA Dou, ZHENG Supei. A Hybrid Scheme of Rotational Flux for Solving 2D Euler Equations[J]. Applied Mathematics and Mechanics, 2021, 42(2): 170-179. doi: 10.21656/1000-0887.410196

Citation:

JIA Dou, ZHENG Supei. A Hybrid Scheme of Rotational Flux for Solving 2D Euler Equations[J]. Applied Mathematics and Mechanics, 2021, 42(2): 170-179. doi: 10.21656/1000-0887.410196

Citation:

JIA Dou, ZHENG Supei. A Hybrid Scheme of Rotational Flux for Solving 2D Euler Equations[J]. Applied Mathematics and Mechanics, 2021, 42(2): 170-179. doi: 10.21656/1000-0887.410196

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A Hybrid Scheme of Rotational Flux for Solving 2D Euler Equations

doi: 10.21656/1000-0887.410196

School of Sciences, Chang’an University, Xi’an 710064, P.R.China

Funds: The National Natural Science Foundation of China（11971075；11401045；11901051）

Received Date: 2020-07-19
Rev Recd Date: 2020-07-19
Publish Date: 2021-02-01

Abstract

Abstract

To improve the resolution of the numerical results for the 2D Euler equations, a hybrid scheme of rotational flux was proposed. The algorithm was built under the quasi-1D idea of the rotational flux method, and the flux function was given in the form of a hybrid scheme coupling the entropy stable numerical flux satisfying the 2nd law of thermodynamics with the HLL numerical flux of good robustness. The time was advanced with the 3rd-order strongly stable Runge-Kutta method. The hybrid scheme of rotational flux has the advantages of simple structure and high resolution. Numerical results show good characteristics of the algorithm.
- rotational invariance,
- entropy stable scheme,
- HLL scheme,
- Euler equation

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References(23)

References

[1]	GODUNOV S. Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics[J]. Matematiceskij Sbornik,1959,47(2): 271-306.
[2]	VAN LEER B.Towards the ultimate conservative difference scheme, Ⅴ: a second-order sequel to Godunov’s method[J]. Journal of Computational Physics1979,32(1): 101-136.
[3]	ROE P. Approximate Riemann solvers, parameter vectors, and difference schemes[J]. Journal of Computational Physics,1981,43(1): 357-372.
[4]	LION M S. Mass flux schemes and connection to shock instability[J]. Journal of Computational Physics,2000,160(2): 623-648.
[5]	PANDOLFI M, AMBROSIO D. Numerical instabilities in upwind methods: analysis and cures for the “carbuncle” phenomenon[J]. Journal of Computational Physics,2001,166(2): 271-301.
[6]	QUIRK J J. A contribution to the great Riemann solver debate[J]. International Journal for Numerical Methods in Fluids,1994,18(6): 550-569.
[7]	SANDERS R, MORANO E, DRUGUET M. Multidimensional dissipation for upwind schemes: stability and applications to gas dynamics[J]. Journal of Computational Physics,1998,145(2): 511-537.
[8]	LEVY D W, POWELL K G, VAN LEER B. Use of a rotated Riemann solver for the two-dimensional Euler equations[J]. Journal of Computational Physics,1993,106(2): 201-214.
[9]	REN Y X. A robust shock-capturing scheme based on rotated Riemann solvers[J]. Computer & Fluids,2003,32(6): 1379-1403.
[10]	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.
[11]	ZHANG F, LIU J. Evaluation of rotated upwind schemes for contact discontinuity and strong shock[J]. Computer & Fluids,2016,134: 11-22.
[12]	刘友琼, 封建湖, 任烔, 等. 求解多维Euler方程的二阶旋转混合型格式[J]. 应用数学和力学, 2014,35(5): 542-553.(LIU Youqiong, FENG Jianhu, REN Tong, et al. Second order rotational hybrid scheme for solving multi-dimensional Euler equation[J]. Applied Mathematics and Mechanics,2014,35(5): 542-553.(in Chinese))
[13]	LAX P D. Weak solutions of nonlinear hyperbolic equations and their numerical computation[J]. Communications on Pure and Applied Mathematics,1954,7(1): 159-193.
[14]	ISMAIL F, ROE P L. Affordable, entropy-consistent Euler flux functions Ⅱ: entropy production at shocks[J]. Journal of Computational Physics,2009,228(4): 5410-5436.
[15]	TADMOR E. Entropy stable approximations of Navier-Stokes equations with no artificial numerical viscosity[J]. Journal of Hyperbolic Differential Equations,2006,3(3): 529-559.
[16]	郑素佩, 王令, 王苗苗. 求解二维浅水波方程的移动网格旋转通量法[J]. 应用数学和力学, 2020,41(1): 42-53.(ZHENG Supei, WANG Ling, WANG Miaomiao. Solution of 2D shallow water wave equation with the moving-grid rotating-invariance method[J]. Applied Mathematics and Mechanics,2020,41(1): 42-53.(in Chinese))
[17]	TORO F. Riemann Solvers and Numerical Methods for Fluid Dynamics [M]. Berlin: Springer, 2013.
[18]	SHU C W. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws[J]. National Aeronautics and Space Administration,1997,97(1): 206-253.
[19]	TADMOR E. The numerical viscosity of entropy stable schemes for systems of conservation laws: Ⅰ[J]. Mathematics of Computation,1987,49(179): 91-103.
[20]	TADMOR E. Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems[J]. Acta Numerica,2003,12: 451-512.
[21]	HARTEN A, LAX P D, VAN LEER B. On upstream differencing and Godunov-type schemes for hyperbolic conservation laws[J]. SIAM Review,1983,25(1): 35-61.
[22]	LAX P, LIU X D. Solution of two-dimensional Riemann problems of gas dynamics by positive schemes[J]. SIAM Journal on Scientific Computing,1998,19(2): 319-340.
[23]	WOODWARD P, COLELLA P. The numerical simulation of two-dimensional fluid flow with strong shocks[J]. Journal of Computation Physics,1984,54(1): 115-173.