二维浅水波方程和欧拉方程数值激波不稳定性分析
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摘要
当使用能够精确分辨接触间断的数值格式来计算多维流体力学问题时,会发现激波附近的扰动有时会出现剧烈增大的现象,这就是所谓的数值激波不稳定性现象。许多文献对二维欧拉方程部分数值格式的不稳定性现象进行了分析和解释,并给出了治愈激波不稳定性的方法。但几乎所有的文献均未讨论过浅水波方程的激波不稳定性问题。本文对二维浅水波方程和欧拉方程数值激波不稳定性问题进行了研究。
在对数值激波不稳定性研究现状进行回顾之后,论文的第一部分研究二维浅水波方程的数值激波不稳定性问题。通过对部分数值格式的线性稳定性分析和相关数值试验,我们发现临界稳定性与部分格式的激波不稳定现象存在密切联系。部分格式在剪切粘性和非线性波粘性的不足,导致了二维浅水波方程的数值激波不稳定性现象。利用临界稳定性分析,文中提出了一种能够消除数值不稳定性现象的混合型数值格式。
论文的第二部分将二维浅水波方程数值激波不稳定性的研究方法推广到二维欧拉方程。与浅水波方程研究不同的是,欧拉方程部分格式的稳定性除了需要足够的剪切粘性,还需要对质量方程进行改进。与之类似的是,通过修改临界稳定的特征值,构造了相应的混合型数值格式来消除二维欧拉方程的数值激波不稳定性现象。
数值试验展示了混合格式对于消除二维浅水波方程和欧拉方程激波数值不稳定现象的有效性和健壮性。
When using the numerical schemes which can accurately capture discontinuity to calculate multidimensional fluid mechanics problems, we will find the perturbation near the shock wave increases dramatically. This is what we call numerical shock instability. Several attempts have been made to understand and cure the instability phenomenon on 2D Euler equations. But few literatures discussed the numerical shock instability of 2D shallow water equations. The nonlinear structure of the multidimensional shallow water equations does not include contact wave but has shear waves. The study of this simple structure provides an insight into the mechanism of shock instability.
In the first part of this paper, we will discuss the numerical shock instability of 2D shallow water wave equations. By analyzing linear stability of some numerical schemes and testifying of some numerical test problems. We find the marginal stability of schemes and dissipation of nonlinear waves result in shock instability of shallow water equations. According to this, we design a hybrid method to remedy the nonphysical phenomenon only by slightly and locally modify the original schemes. The numerical experiments prove efficiency and robustness of hybrid scheme to eliminate shock instability of 2D shallow water equations.
In the second part of this paper, we will apply the analysis method of 2D shallow water equations to 2D Euler equations, and devise corresponding hybrid scheme to remedy shock instability of 2D Euler equations.
在对数值激波不稳定性研究现状进行回顾之后,论文的第一部分研究二维浅水波方程的数值激波不稳定性问题。通过对部分数值格式的线性稳定性分析和相关数值试验,我们发现临界稳定性与部分格式的激波不稳定现象存在密切联系。部分格式在剪切粘性和非线性波粘性的不足,导致了二维浅水波方程的数值激波不稳定性现象。利用临界稳定性分析,文中提出了一种能够消除数值不稳定性现象的混合型数值格式。
论文的第二部分将二维浅水波方程数值激波不稳定性的研究方法推广到二维欧拉方程。与浅水波方程研究不同的是,欧拉方程部分格式的稳定性除了需要足够的剪切粘性,还需要对质量方程进行改进。与之类似的是,通过修改临界稳定的特征值,构造了相应的混合型数值格式来消除二维欧拉方程的数值激波不稳定性现象。
数值试验展示了混合格式对于消除二维浅水波方程和欧拉方程激波数值不稳定现象的有效性和健壮性。
When using the numerical schemes which can accurately capture discontinuity to calculate multidimensional fluid mechanics problems, we will find the perturbation near the shock wave increases dramatically. This is what we call numerical shock instability. Several attempts have been made to understand and cure the instability phenomenon on 2D Euler equations. But few literatures discussed the numerical shock instability of 2D shallow water equations. The nonlinear structure of the multidimensional shallow water equations does not include contact wave but has shear waves. The study of this simple structure provides an insight into the mechanism of shock instability.
In the first part of this paper, we will discuss the numerical shock instability of 2D shallow water wave equations. By analyzing linear stability of some numerical schemes and testifying of some numerical test problems. We find the marginal stability of schemes and dissipation of nonlinear waves result in shock instability of shallow water equations. According to this, we design a hybrid method to remedy the nonphysical phenomenon only by slightly and locally modify the original schemes. The numerical experiments prove efficiency and robustness of hybrid scheme to eliminate shock instability of 2D shallow water equations.
In the second part of this paper, we will apply the analysis method of 2D shallow water equations to 2D Euler equations, and devise corresponding hybrid scheme to remedy shock instability of 2D Euler equations.
引文
[1]J. L. Steger, R. F. Warming, Flux vector-splitting of the inviscid gas dynamics equations with application to finite difference methods[J], Journal of Computational Physics,1981;40:263-293.
[2]B. van Leer, Flux-vector splitting for the Euler equations [J], Technical Report ICASE 82-30, NASA Langley Research Center, USA,1982;
[3]M. S. Liou, C. J. Steffen, A new flux splitting scheme [J], Journal of Computational Physics,1993:107:23-39.
[4]S. K. Godunov, A finite difference method for computation of discontinuous solutions of the equations of fluid dynamics[J], Mat. Sb.,1959; 47:271-306.
[5]P. L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes [J], Journal of Computational Physics,1981:43;357-372.
[6]P. L. Roe, J. Pike, Efficient Construction and Utilisation of Approximate Riemann solutions [J], In Computing Methods in Applied Science and Engineering. North-Holland,1984.
[7]A. Harten, P. D. Lax, B. Van Leer, On Upstream Differencing and Godunov Type Schemes for Hyperbolic Conservation Laws[J], SIAM Review,1983:25;35-61.
[8]E. F. Toro, M. Spruce, W. Speares, Restoration of the Contact Surface in the HLL-Riemann solver[J], Shock Waves,1994:4;25-34.
[9]M. S. Liou, Recent Progress and Applications of AUSM+[A], In 16th international conference on numerical methods in Fluid Dynamics, Lecture Notes in Physics 515, Springer-Verlag,1998;302-307.
[10]A. Harten, High resolution schemes for hyperbolic conservation laws[J], Journal of Computational Physics,1983:49;357-393.
[11]P. K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws[J], SIAM Journal of numerical analysis,1984:21(5); 995-1011.
[12]H. C. Yee, Construction of explicit and implicit symmetric TVD schemes and their applications[J], Journal of Computational Physics,1987:68;151-179.
[13]B. Van Leer, Towards the ultimate conservation difference scheme V, A second-order sequel to Godunov's method [J], Journal of Computational Physics,1979:32; 101-136.
[14]P. R. Woodward, P. Colella, The numerical simulation of two-dimensional fluid flow with strong shocks[J], Journal of Computational Physics,1984:54;115-173.
[15]A. Harten, B. Engquist, S. Osher, R. Chakravarthy, Uniformly high-order accurate essentially non-oscillatory schemes[J], Journal of Computational Physics, 1987:71;231-303.
[16]X. D. Liu, S. Osher, T. Chan, Weighted essentially non-oscillatory schemes [J], Journal of Computational Physics,1994:115:200-212.
[17]K. M. Perry, S. T. Imlay, Blunt Body Flow Simulations[A], AIAA Paper,1988; 88-2924.
[18]J. Quirk, A contribution to the Great Riemann Solver Debate[J], International Journal Numerical Method Fluid,1994:18:555-574.
[19]Hao Wu, Longjun Shen, Zhijun Shen, A Hybrid Numerical Method to Cure Numerical Shock Instability[J], Communications in Computational Physics,2010:8(5); 1264-1271.
[20]M. S. Liou, Mass Flux Scheme and Connection to Shock Instability[J], Journal of Computational Physics,2000:160:623-648.
[21]J. Gressier, J. M. Moschetta, Robustness versus Accuracy in Shock-Wave Computations [J], International Journal of Numerical Methods in Fluids,2000:33; 313-332.
[22]D. I. Pullin, Direct Simulation Methods for Compressible Inviscid Ideal Gas Flow[J], Journal Computational Physics,1980:34:231-244.
[23]M. N. Macrossan, R. I. Oliver, A Kinetic Theory Solution Method for the Navier-Stokes Equations[J], International Journal of Numerical Methods in Fluids,1993:17:177-193.
[24]K. Xu, Gas-kinetic schemes for unsteady compressible flow simulations [J], VKI LS,1998:03:1-10.
[25]K. Xu, Z. Li, Dissipative Mechanismin Godunov-type Schemes[J], International Journal of Numerical Methods in Fluids,2001:37;1-22.
[26]S. Kim, C. Kim,O.-H Rho, S. K. H, Cures for the shock instability:Development of a shock-stable Roe scheme[J], Journal of Computational Physics,2003:185; 342-374.
[27]M. Dumbser, J. M. Moschetta, J. Gressier, A matrix stability analysis of the carbuncle phenomenon[J], Journal of Computational Physics,2004:197(2); 647-670.
[28]Y. Wada, M. S. Liou, An Accurate and Robust Flux Splitting Scheme for Shock and Contact Discontinuities[J], SIAM Journal of Scientific Computation,1997: 18(3); 633-657.
[29]J. M. Moschetta, J. Gressier, J. C. Robinet, G. Casalis, The Carbuncle Phenomenon:a Genuine Euler Instability?, in Godunov Methods, Theory and Applications, Ed. E. F. Toro, Kluwer Academic/Plenum Publ.,1995;639-645.
[30]H. C. Lin, Dissipation addition to flux-difference splitting[J], Journal of Computational Physics,1995:117;20-25.
[31]K. Sermeus, H. Deconinck, Solution of steady Euler and Navier-Stokes equations using Residual Distribution schemes[A],33rd Computational Fluid Dynamics Course, VKI Lecture Series2003-05(2003).
[32]M. Pandolfi, D. D'Ambrosio, Numerical Instabilities in Upwind Methods: Analysis And Cures for the "Carbuncle" Phenomenon[J], Journal of Computational Physics,2001:166;271-301.
[33]S. Davis, A rotationally biased upwind difference scheme for the Euler equations[J], Journal Computational Physics,1984:56;65-92.
[34]D. W. Levy, K. G. Powell, B. Van Leer, Use of a rotated Riemann Solver for the two-dimensional Euler equations [J], Journal of Computational Physics,1993:106; 204-214.
[35]Y. X. Ren, A robust shock-capturing scheme based on rotated Riemann solvers[J], Computers & Fluids,2003:32;1379-1403.
[36]Nishikawa, Kitamura, Very simple, carbuncle-free, boundary-layer-resolving, rotated-hybrid Riemann solvers [J], Journal of Computational Physics,2008:227; 2560-2581.
[37]S. F. Davis, Simplified Second-Order Godunov-Type Methods [J], SIAM J. Sci. Stat. Comput.,1988:9;445-473.
[38]B. Einfeldt, On Godunov-Type Methods for Gas Dynamics [J], SIAM J. Numer. Anal., 1988:25(2);294-318.
[39]E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics[M], Springer,1999; 315-591.
[40]王继海,二维非定常流和激波[M],北京：科学出版社,第一版,1994。
[41]D. I. Blokhintzev, The driven receiver of sound waves, Dokl. Akad. Nauk SSSR 47, 22 (1945).
[42]J. Griffond, Linear interaction analysis applied to a mixture of two perfect Gases[J], Phys. Fluids,2005:17;086-101.
[2]B. van Leer, Flux-vector splitting for the Euler equations [J], Technical Report ICASE 82-30, NASA Langley Research Center, USA,1982;
[3]M. S. Liou, C. J. Steffen, A new flux splitting scheme [J], Journal of Computational Physics,1993:107:23-39.
[4]S. K. Godunov, A finite difference method for computation of discontinuous solutions of the equations of fluid dynamics[J], Mat. Sb.,1959; 47:271-306.
[5]P. L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes [J], Journal of Computational Physics,1981:43;357-372.
[6]P. L. Roe, J. Pike, Efficient Construction and Utilisation of Approximate Riemann solutions [J], In Computing Methods in Applied Science and Engineering. North-Holland,1984.
[7]A. Harten, P. D. Lax, B. Van Leer, On Upstream Differencing and Godunov Type Schemes for Hyperbolic Conservation Laws[J], SIAM Review,1983:25;35-61.
[8]E. F. Toro, M. Spruce, W. Speares, Restoration of the Contact Surface in the HLL-Riemann solver[J], Shock Waves,1994:4;25-34.
[9]M. S. Liou, Recent Progress and Applications of AUSM+[A], In 16th international conference on numerical methods in Fluid Dynamics, Lecture Notes in Physics 515, Springer-Verlag,1998;302-307.
[10]A. Harten, High resolution schemes for hyperbolic conservation laws[J], Journal of Computational Physics,1983:49;357-393.
[11]P. K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws[J], SIAM Journal of numerical analysis,1984:21(5); 995-1011.
[12]H. C. Yee, Construction of explicit and implicit symmetric TVD schemes and their applications[J], Journal of Computational Physics,1987:68;151-179.
[13]B. Van Leer, Towards the ultimate conservation difference scheme V, A second-order sequel to Godunov's method [J], Journal of Computational Physics,1979:32; 101-136.
[14]P. R. Woodward, P. Colella, The numerical simulation of two-dimensional fluid flow with strong shocks[J], Journal of Computational Physics,1984:54;115-173.
[15]A. Harten, B. Engquist, S. Osher, R. Chakravarthy, Uniformly high-order accurate essentially non-oscillatory schemes[J], Journal of Computational Physics, 1987:71;231-303.
[16]X. D. Liu, S. Osher, T. Chan, Weighted essentially non-oscillatory schemes [J], Journal of Computational Physics,1994:115:200-212.
[17]K. M. Perry, S. T. Imlay, Blunt Body Flow Simulations[A], AIAA Paper,1988; 88-2924.
[18]J. Quirk, A contribution to the Great Riemann Solver Debate[J], International Journal Numerical Method Fluid,1994:18:555-574.
[19]Hao Wu, Longjun Shen, Zhijun Shen, A Hybrid Numerical Method to Cure Numerical Shock Instability[J], Communications in Computational Physics,2010:8(5); 1264-1271.
[20]M. S. Liou, Mass Flux Scheme and Connection to Shock Instability[J], Journal of Computational Physics,2000:160:623-648.
[21]J. Gressier, J. M. Moschetta, Robustness versus Accuracy in Shock-Wave Computations [J], International Journal of Numerical Methods in Fluids,2000:33; 313-332.
[22]D. I. Pullin, Direct Simulation Methods for Compressible Inviscid Ideal Gas Flow[J], Journal Computational Physics,1980:34:231-244.
[23]M. N. Macrossan, R. I. Oliver, A Kinetic Theory Solution Method for the Navier-Stokes Equations[J], International Journal of Numerical Methods in Fluids,1993:17:177-193.
[24]K. Xu, Gas-kinetic schemes for unsteady compressible flow simulations [J], VKI LS,1998:03:1-10.
[25]K. Xu, Z. Li, Dissipative Mechanismin Godunov-type Schemes[J], International Journal of Numerical Methods in Fluids,2001:37;1-22.
[26]S. Kim, C. Kim,O.-H Rho, S. K. H, Cures for the shock instability:Development of a shock-stable Roe scheme[J], Journal of Computational Physics,2003:185; 342-374.
[27]M. Dumbser, J. M. Moschetta, J. Gressier, A matrix stability analysis of the carbuncle phenomenon[J], Journal of Computational Physics,2004:197(2); 647-670.
[28]Y. Wada, M. S. Liou, An Accurate and Robust Flux Splitting Scheme for Shock and Contact Discontinuities[J], SIAM Journal of Scientific Computation,1997: 18(3); 633-657.
[29]J. M. Moschetta, J. Gressier, J. C. Robinet, G. Casalis, The Carbuncle Phenomenon:a Genuine Euler Instability?, in Godunov Methods, Theory and Applications, Ed. E. F. Toro, Kluwer Academic/Plenum Publ.,1995;639-645.
[30]H. C. Lin, Dissipation addition to flux-difference splitting[J], Journal of Computational Physics,1995:117;20-25.
[31]K. Sermeus, H. Deconinck, Solution of steady Euler and Navier-Stokes equations using Residual Distribution schemes[A],33rd Computational Fluid Dynamics Course, VKI Lecture Series2003-05(2003).
[32]M. Pandolfi, D. D'Ambrosio, Numerical Instabilities in Upwind Methods: Analysis And Cures for the "Carbuncle" Phenomenon[J], Journal of Computational Physics,2001:166;271-301.
[33]S. Davis, A rotationally biased upwind difference scheme for the Euler equations[J], Journal Computational Physics,1984:56;65-92.
[34]D. W. Levy, K. G. Powell, B. Van Leer, Use of a rotated Riemann Solver for the two-dimensional Euler equations [J], Journal of Computational Physics,1993:106; 204-214.
[35]Y. X. Ren, A robust shock-capturing scheme based on rotated Riemann solvers[J], Computers & Fluids,2003:32;1379-1403.
[36]Nishikawa, Kitamura, Very simple, carbuncle-free, boundary-layer-resolving, rotated-hybrid Riemann solvers [J], Journal of Computational Physics,2008:227; 2560-2581.
[37]S. F. Davis, Simplified Second-Order Godunov-Type Methods [J], SIAM J. Sci. Stat. Comput.,1988:9;445-473.
[38]B. Einfeldt, On Godunov-Type Methods for Gas Dynamics [J], SIAM J. Numer. Anal., 1988:25(2);294-318.
[39]E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics[M], Springer,1999; 315-591.
[40]王继海,二维非定常流和激波[M],北京：科学出版社,第一版,1994。
[41]D. I. Blokhintzev, The driven receiver of sound waves, Dokl. Akad. Nauk SSSR 47, 22 (1945).
[42]J. Griffond, Linear interaction analysis applied to a mixture of two perfect Gases[J], Phys. Fluids,2005:17;086-101.