一个新的Riemann解法器及相关问题的研究
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摘要
本文研究双曲守恒律方程的Godunov型格式。全文分为两部分。第一部分提出了一个新的Riemann解法器，利用两步分裂的思想，对原Riemann问题的求解转化为两个相关的子问题的求解。这种方法可以避开避开传统Riemann解法器中复杂的波系分解和通量选择，较为方便地构造一个原Riemann问题的近似解。将这个新Riemann解法器写成等价的数值通量形式，可以发现该格式是单调的，利用Harten定理证明了该格式是TVD的，并证明了该格式满足离散熵条件。大量的数值试验表明该Riemann解法器的健壮性好，应用范围广泛。
第二部分研究了Burgers方程和Euler方程数值计算中的声速点故障问题。对数值格式产生该现象的原因进行了分析。应用前述的两步分裂的思想分解原方程，通过增加或减少线性对流项使得每步的子问题中都不包括跨声速区，从而避开了声速点故障现象。将此方法实施于一系列典型格式，均取得了令人满意的结果。
This paper consists of two parts. In the first part, the Riemann solvers in Godunov-type schemes are considered in the hyperbolic conservation laws systems. Based on the two-step splitting method, a new approximate Riemann solver is presented. In the first step we add a linear advection term to the equations of hyperbolic conservation laws to make the Godunov numerical flux in the interface between cells is computed easily, then the added linear advection term is thrown off in the second step. During the computing procedure, the algorithm does not need iterative technique and characteristic wave decomposition. After the new solver is expressed a form of a new numerical flux, the new scheme is proved monotonic and TVD and satisfies the entropy condition. A lot of numerical results show its robustness and the solver can be applied widely.
In the second part, we show reasons that the sonic point glitch in the computation of conservation law happens. A new method is presented to cure the sonic glitch. The method that based on the two-step splitting method is similar to the present solver presented in the first part. Applying this method to many schemes with sonic point glitch, the sonic glitch is cured.
第二部分研究了Burgers方程和Euler方程数值计算中的声速点故障问题。对数值格式产生该现象的原因进行了分析。应用前述的两步分裂的思想分解原方程，通过增加或减少线性对流项使得每步的子问题中都不包括跨声速区，从而避开了声速点故障现象。将此方法实施于一系列典型格式，均取得了令人满意的结果。
This paper consists of two parts. In the first part, the Riemann solvers in Godunov-type schemes are considered in the hyperbolic conservation laws systems. Based on the two-step splitting method, a new approximate Riemann solver is presented. In the first step we add a linear advection term to the equations of hyperbolic conservation laws to make the Godunov numerical flux in the interface between cells is computed easily, then the added linear advection term is thrown off in the second step. During the computing procedure, the algorithm does not need iterative technique and characteristic wave decomposition. After the new solver is expressed a form of a new numerical flux, the new scheme is proved monotonic and TVD and satisfies the entropy condition. A lot of numerical results show its robustness and the solver can be applied widely.
In the second part, we show reasons that the sonic point glitch in the computation of conservation law happens. A new method is presented to cure the sonic glitch. The method that based on the two-step splitting method is similar to the present solver presented in the first part. Applying this method to many schemes with sonic point glitch, the sonic glitch is cured.
引文
[1] A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys, 1983; 49: 357-393.
[2] P. K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws. SIAMJ. Nume. Ana, 21(1984): 995-1011.
[3] H. C. Yee, Construction of explicit and implicit symmetric TVD schemes and their applications, J. Comput Phys, 68(1987): 151-179.
[4] S. K. Godunov, A finite difference method for computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sb, 47(1959):271-306.
[5] B. Van Leer, Towards the ultimate conservation difference scheme V, A second-order sequel to Godunov's method . J. Comput Phys, 32(1979): 101-136.
[6] P. R. Woodward and P. Colella, The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput Phys, 54(1984): 115-173.
[7] A. Harten, B. Engquist, S. Osher and R. Chakravarthy, Uniformly high-order accurate essentially non-oscillatory schemes, J. Comput. Phys, 71(1987): 231-303.
[8] C. W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput Phys, 77(1988): 439-471.
[9] C. W. Shu and S. Osher. Efficient implementation of essentially non-oscillatory shock capturing schemes II, J. Comput Phys, 83(1989): 32-78.
[10] X. D. Liu, S. Osher and T. Chan, Weighted essentially non-oscillatory schemes. J. Comput Phys, 115(1994): 200-212.
[11] P. L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys, 43(1981)357-372.
[12] P. L. Roe and J. Pike. Efficient construction and utilisation of approximate Riemann Solutions. In Computing Methods in Applied Science and Engineering. North-Holland, 1984.
[13] A. Harten and J. M. Hyman, Self-adjusting grid methods for one-dimensional hyperbolic conservation laws, J. Comput. Phys. 50(1983): 235-269.
[14] A. Harten, P. D. Lax and B. Van Leer. On upstream differencing and Godunov-type schemes for Hyperbolic Conservation Laws. SIAM Review, 25(1983):35-61.
[15] E. F. Toro, M. Spruce and W. Speares, Restoration of the contact surface in the HLL-Riemann solver. Shock Waves, 4(1994):25-34.
[16] E. F. Toro, Direct Riemann Solver for the time-dependent Euler equations. Shock Waves, 5(1995):75-80.
[17] B. Engquist and S. Osher, One sided difference approximations for nonlinear conservation laws, Math. Comp, 36 (1981):321-351.
[18] E. F. Toro, Riemann solver and numerical methods for fluid dynamics, second. Springer, Berlin, 1999.
[19] J. L. Steger, R. F. Warming, Flux vector-splitting of the inviscid gas dynamics equations with application to finite difference methods, J. Comput. Phys. 40(1981)263-293.
[20] B. van Leer, Flux-vector splitting for the Euler equations, Technical Report ICASE 82-30, NASA Langley Research Center, USA, 1982.
[21] M. S. Liou, C. J. Steffen, A new flux splitting scheme, J. Comput. Phys. 107(1993)23-39.
[22] J. L. Steger, R. F. Warming, Flux vector-splitting of the inviscid gas dynamics equations with application to finite difference methods, J. Comput. Phys. 40(1981)263-293.
[23] B. van Leer, W. T. Lee and K. G. Powell, Sonic-point-capturing, AIAA-89-1945-CP, in AIAA 9~(th) Computational Fluid Dynamics Conference, Buffalo, NY, 1989.
[24] P. L. Roe, Sonic flux formulae, SIAM J. Sci. Stat. Comput. 13(1992):611-630.
[25] M. J. Kermani and E. G. Plett, Modified entropy correction formula for the Roe scheme, American Institute of Aeronautics and Astronautics Paper 2001-0083.
[26] F. J. Liu and W. W. Liou, A new approach for eliminating numerical oscillations of Roe family of schemes at sonic point. AIAA99-0301, in: 37~(th) AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, 1999.
[27] J. M. Moschetta and J. Gressier, The sonic point glitch problems: a numerical solution, in Charles-Henri Bruneau(Ed.) 16th International Conference on Numerical Method in Fluid Dynamics, 1998, pp403-408.
[28] J. M. Moschetta and J. Gressier, A cure for the sonic point glitch, Int. J. Comput, Fluid. Dynamics, 13(2000): 143-159.
[29] H. Z. Tang and K. Xu, An explanation for the sonic point glitch, preprint, 2000.
[30] H. Z. Tang, On the sonic point glitch. J. Comput. Phys. 202(2005) 507-532.
[31] 水鸿寿，一维流体力学差分方法．北京：国防工业出版社，1998。
[32] 应隆安、滕振寰，双曲形守恒律方程及其差分方法。科学出版社，199
[33] Edwige Godlewski and Pireer-Arnaud Raviart, Numerical approximation of hyperbolic systems of conservation laws, Spinger, 1996.
[34] P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves. SIAM Regional Conf Series Lectures in Appl Math, 1972,11.
[35] M. J. Ivings, D. M. Causon and E. F. Toro, On Riemann solvers for compressible liquids, Int. J. Nume. Meth. Fluids, 28(1998)392-418.
[36] J. E. Jackson and M. A. Jamnia, Non-linear FSI due to underwater explosions, J. Eng. Mech. 110(1984)507-517.
[37] Buffard, T. Gallouet and J. M. Herard, A sequel to a rough Godunov scheme: Application to real gases. Comput. Fluids, 29(2000): 813-847.
[38] T. Gallouet, J. M. Herard and N. Seguin, Some recent finite volume schemes to compute Euler equations using real gas EOS. Int. J. Numer. Meth. Fluids, 2000.
[39] 刘儒勋、舒其望，计算流体力学的若干新方法。北京：科学出版社，2003。
[40] 谭维炎，计算浅水动力学：有限体积法的应用。北京：清华大学出版社，1998。
[41] 李大潜、秦铁虎，物理学与偏微分方程。北京：高等教育出版社，2005。
[42] Brio, M. and Wu. C. C, An upwind differencing scheme for the Equations of ideal magnetohydrodynamics, J. Comp. Phys, 75(1988): 400-422.
[43] J. Von Neumann and R. D. Richtmyer. A method for the numerical calculation of hydrodynamic shocks. Journal of Apply physics, 21 (1951): 232-237.
[44] V. V. Rusanov, Calculation of interaction of non-steady shock waves with obstacles, J. Comp. Math. Phys, USSR, 1(1961): 267-279.
[45] J. K. Dukowica, A general non-iterative Riemann solver for Godunov's method. J. Comp. Phys, 31(1985): 119-137.
[46] J. Pike, Riemann solver for perfect and near-perfect gases. AIAA Journal, 31(1993): 1801-1808.
[47] S. Osher and F. Solomon, Upwind difference schemes for hyperbolic conservation laws, Math Comp, 38(1982): 339-374.
[48] S. Osher and S. Chakravarthy, Upwind schemes and boundary conditions with applications to Euler equations in general Geometries, J. Comput. Phys, 50(1983):447-481.
[49] S. F. Davis, Simplified second-order Godunov-type methods, SIAM J. Sci. Stat. Comp, 9(1988):445-473.
[50] B. Einfeldt, On Godunov-type methods for gas dynamics. SIAM, J.Numer. Anal. 25(1988)29-318.
[51] B. Van. Leer, Towards the ultimate conservative difference scheme V:A second order sequel to Godunov's method, J. Comput. Phys, 32(1979)101-136.
[52] Chi-wang Shu and Guang-shan Jiang, Efficient implementation of weighted ENO Schemes, J. Comput. Phys., 126(1996):202-228.
[53] 傅德薰、马延文,计算流体力学。北京:高等教育出版社, 2002.
[54] P. D. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical computation, Communication in Pure and Applied Mathematics, 7(1954): 159.
[55] Vincent Guinot, Godunov-type schemes: an introduction for engineers, Elsevier, 2003.
[56] J. U. Brackbill and D. C. Barnes, The effect of nonzero V·B on the numerical solution of the magnetohydrodynamic equations, J. Comput. Phys, 35(1980): 426-430.
[57] W. Dai and P. R. Woodward, A simple finite difference scheme for multidimensional magnetohydrodynamics, J. Comput. Phys, 142(1998): 331-369.
[58] Gabor Toth, The ▽·B = 0 constraint in shock-capturing magnetohydrodynamics, J. Comput. Phys, 161(2000):605-652.
[59] P. Janhunen, A positive conservative method for magnetohydrodynamics based on HLL and Roe methods, J. Comput. Phys, 160(2000): 649-661.
[60] K. G. Powell, P. L. Roe, T. J. Linde, T. I. Gombosi and D. L. Dezeeuw, A solution-adaptive upwinding scheme for ideal magnetohydrodynamics, J. Comput. Phys, 154(1999): 284.
[61] J. A. Sod, A Survey of Several Finite Difference Methods for Systems of Nonlinear Hyperbolic Conservation Laws, J. Comput. Phys., 27(1978): 1-31.
[62] P. Woodward and P. Colella, The Numerical Simulation of Two-Dimensional Fluid Flow with Strong Shocks, J. Comput. Phys, 54(1984):115-173.
[63] T. W. Roberts, The Behavior of Flux Difference Splitting Schemes near Slowly Moving Shock Waves, J. Comput. Phys, 10(1990):141-160.
[64] A. Kurganov and E. Tadmor, Solution of two-dimensional Riemann problem for gas dynamics without Riemann problem solvers.
[65] C. W. Schulz-Rinne, J. P. Collins and H. M. Glaz, Numerical solution of the Riemann problem for two-dimensional gas dynamics, SIAM J. Sci. Comput, 14(1993): 1394-1414.
[66] P. D. Lax and Xu-Dong Liu, Solution of the Riemann problem of two-dimensional gas dynamics by positive schemes, SIAM J. Sci. Comput, 19(1998):319-340.
[67] C. W. Schulz-Rinne, Classification of the Riemann problem for two-dimensional gas dynamics, SIAM J. Math. Anal, 24(1993):76-88.
[68] N.N. Kruzkov, First order quasilinear equations in several independent variables, Math. USSR Sbornik, 10(1970):217-243.
[69] J. P. Vila, High-order schemes and entropy condition for nonlinear hyperbolic systems of conservation laws, Mathematics of computation, 50(1986)53-73.
[70] A.Harten, On the symmetric form of systems of conservation laws with entropy, J. Comput. Phys, 49(1983):151-164.
[71] S. Osher, Riemann solver, the entropy condition, and difference approximations, SIAM J. Num. Anal, 21(1984):217-235.
[72] S. Osher and A. Majda, Numerical viscosity and the entropy condition, Communications on Pure and Applied Mathmatics, (1979):797-838.
[73] S. Osher and S. Chakravarthy, High resolution schemes and the entropy condition, SIAM J.Num.Anal, 21(1984):955-984.
[74] T. Zhang and Y. Zheng, Conjecture on the structure on the solutions of the Riemann problem for two-dimensional gas dynamics systems, SIAM J. Math. Anal, 21(1990):593-630.
[75] T. Chang, G. Q. Chen and S. Yang, On the 2-D Riemann problem for the compressible Euler equations, I. Interaction of shock and rarefaction waves, Discrete Contin. Dynam. Systems, l(1995):555-584.
[76] T. Chang, G. Q. Chen and S. Yang, On the 2-D Riemann problem for the compressible Euler equations, II. Interaction of contact discontinuities, Discrete Contin. Dynam. Systems, 6(2000) :419-430.
[2] P. K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws. SIAMJ. Nume. Ana, 21(1984): 995-1011.
[3] H. C. Yee, Construction of explicit and implicit symmetric TVD schemes and their applications, J. Comput Phys, 68(1987): 151-179.
[4] S. K. Godunov, A finite difference method for computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sb, 47(1959):271-306.
[5] B. Van Leer, Towards the ultimate conservation difference scheme V, A second-order sequel to Godunov's method . J. Comput Phys, 32(1979): 101-136.
[6] P. R. Woodward and P. Colella, The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput Phys, 54(1984): 115-173.
[7] A. Harten, B. Engquist, S. Osher and R. Chakravarthy, Uniformly high-order accurate essentially non-oscillatory schemes, J. Comput. Phys, 71(1987): 231-303.
[8] C. W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput Phys, 77(1988): 439-471.
[9] C. W. Shu and S. Osher. Efficient implementation of essentially non-oscillatory shock capturing schemes II, J. Comput Phys, 83(1989): 32-78.
[10] X. D. Liu, S. Osher and T. Chan, Weighted essentially non-oscillatory schemes. J. Comput Phys, 115(1994): 200-212.
[11] P. L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys, 43(1981)357-372.
[12] P. L. Roe and J. Pike. Efficient construction and utilisation of approximate Riemann Solutions. In Computing Methods in Applied Science and Engineering. North-Holland, 1984.
[13] A. Harten and J. M. Hyman, Self-adjusting grid methods for one-dimensional hyperbolic conservation laws, J. Comput. Phys. 50(1983): 235-269.
[14] A. Harten, P. D. Lax and B. Van Leer. On upstream differencing and Godunov-type schemes for Hyperbolic Conservation Laws. SIAM Review, 25(1983):35-61.
[15] E. F. Toro, M. Spruce and W. Speares, Restoration of the contact surface in the HLL-Riemann solver. Shock Waves, 4(1994):25-34.
[16] E. F. Toro, Direct Riemann Solver for the time-dependent Euler equations. Shock Waves, 5(1995):75-80.
[17] B. Engquist and S. Osher, One sided difference approximations for nonlinear conservation laws, Math. Comp, 36 (1981):321-351.
[18] E. F. Toro, Riemann solver and numerical methods for fluid dynamics, second. Springer, Berlin, 1999.
[19] J. L. Steger, R. F. Warming, Flux vector-splitting of the inviscid gas dynamics equations with application to finite difference methods, J. Comput. Phys. 40(1981)263-293.
[20] B. van Leer, Flux-vector splitting for the Euler equations, Technical Report ICASE 82-30, NASA Langley Research Center, USA, 1982.
[21] M. S. Liou, C. J. Steffen, A new flux splitting scheme, J. Comput. Phys. 107(1993)23-39.
[22] J. L. Steger, R. F. Warming, Flux vector-splitting of the inviscid gas dynamics equations with application to finite difference methods, J. Comput. Phys. 40(1981)263-293.
[23] B. van Leer, W. T. Lee and K. G. Powell, Sonic-point-capturing, AIAA-89-1945-CP, in AIAA 9~(th) Computational Fluid Dynamics Conference, Buffalo, NY, 1989.
[24] P. L. Roe, Sonic flux formulae, SIAM J. Sci. Stat. Comput. 13(1992):611-630.
[25] M. J. Kermani and E. G. Plett, Modified entropy correction formula for the Roe scheme, American Institute of Aeronautics and Astronautics Paper 2001-0083.
[26] F. J. Liu and W. W. Liou, A new approach for eliminating numerical oscillations of Roe family of schemes at sonic point. AIAA99-0301, in: 37~(th) AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, 1999.
[27] J. M. Moschetta and J. Gressier, The sonic point glitch problems: a numerical solution, in Charles-Henri Bruneau(Ed.) 16th International Conference on Numerical Method in Fluid Dynamics, 1998, pp403-408.
[28] J. M. Moschetta and J. Gressier, A cure for the sonic point glitch, Int. J. Comput, Fluid. Dynamics, 13(2000): 143-159.
[29] H. Z. Tang and K. Xu, An explanation for the sonic point glitch, preprint, 2000.
[30] H. Z. Tang, On the sonic point glitch. J. Comput. Phys. 202(2005) 507-532.
[31] 水鸿寿，一维流体力学差分方法．北京：国防工业出版社，1998。
[32] 应隆安、滕振寰，双曲形守恒律方程及其差分方法。科学出版社，199
[33] Edwige Godlewski and Pireer-Arnaud Raviart, Numerical approximation of hyperbolic systems of conservation laws, Spinger, 1996.
[34] P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves. SIAM Regional Conf Series Lectures in Appl Math, 1972,11.
[35] M. J. Ivings, D. M. Causon and E. F. Toro, On Riemann solvers for compressible liquids, Int. J. Nume. Meth. Fluids, 28(1998)392-418.
[36] J. E. Jackson and M. A. Jamnia, Non-linear FSI due to underwater explosions, J. Eng. Mech. 110(1984)507-517.
[37] Buffard, T. Gallouet and J. M. Herard, A sequel to a rough Godunov scheme: Application to real gases. Comput. Fluids, 29(2000): 813-847.
[38] T. Gallouet, J. M. Herard and N. Seguin, Some recent finite volume schemes to compute Euler equations using real gas EOS. Int. J. Numer. Meth. Fluids, 2000.
[39] 刘儒勋、舒其望，计算流体力学的若干新方法。北京：科学出版社，2003。
[40] 谭维炎，计算浅水动力学：有限体积法的应用。北京：清华大学出版社，1998。
[41] 李大潜、秦铁虎，物理学与偏微分方程。北京：高等教育出版社，2005。
[42] Brio, M. and Wu. C. C, An upwind differencing scheme for the Equations of ideal magnetohydrodynamics, J. Comp. Phys, 75(1988): 400-422.
[43] J. Von Neumann and R. D. Richtmyer. A method for the numerical calculation of hydrodynamic shocks. Journal of Apply physics, 21 (1951): 232-237.
[44] V. V. Rusanov, Calculation of interaction of non-steady shock waves with obstacles, J. Comp. Math. Phys, USSR, 1(1961): 267-279.
[45] J. K. Dukowica, A general non-iterative Riemann solver for Godunov's method. J. Comp. Phys, 31(1985): 119-137.
[46] J. Pike, Riemann solver for perfect and near-perfect gases. AIAA Journal, 31(1993): 1801-1808.
[47] S. Osher and F. Solomon, Upwind difference schemes for hyperbolic conservation laws, Math Comp, 38(1982): 339-374.
[48] S. Osher and S. Chakravarthy, Upwind schemes and boundary conditions with applications to Euler equations in general Geometries, J. Comput. Phys, 50(1983):447-481.
[49] S. F. Davis, Simplified second-order Godunov-type methods, SIAM J. Sci. Stat. Comp, 9(1988):445-473.
[50] B. Einfeldt, On Godunov-type methods for gas dynamics. SIAM, J.Numer. Anal. 25(1988)29-318.
[51] B. Van. Leer, Towards the ultimate conservative difference scheme V:A second order sequel to Godunov's method, J. Comput. Phys, 32(1979)101-136.
[52] Chi-wang Shu and Guang-shan Jiang, Efficient implementation of weighted ENO Schemes, J. Comput. Phys., 126(1996):202-228.
[53] 傅德薰、马延文,计算流体力学。北京:高等教育出版社, 2002.
[54] P. D. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical computation, Communication in Pure and Applied Mathematics, 7(1954): 159.
[55] Vincent Guinot, Godunov-type schemes: an introduction for engineers, Elsevier, 2003.
[56] J. U. Brackbill and D. C. Barnes, The effect of nonzero V·B on the numerical solution of the magnetohydrodynamic equations, J. Comput. Phys, 35(1980): 426-430.
[57] W. Dai and P. R. Woodward, A simple finite difference scheme for multidimensional magnetohydrodynamics, J. Comput. Phys, 142(1998): 331-369.
[58] Gabor Toth, The ▽·B = 0 constraint in shock-capturing magnetohydrodynamics, J. Comput. Phys, 161(2000):605-652.
[59] P. Janhunen, A positive conservative method for magnetohydrodynamics based on HLL and Roe methods, J. Comput. Phys, 160(2000): 649-661.
[60] K. G. Powell, P. L. Roe, T. J. Linde, T. I. Gombosi and D. L. Dezeeuw, A solution-adaptive upwinding scheme for ideal magnetohydrodynamics, J. Comput. Phys, 154(1999): 284.
[61] J. A. Sod, A Survey of Several Finite Difference Methods for Systems of Nonlinear Hyperbolic Conservation Laws, J. Comput. Phys., 27(1978): 1-31.
[62] P. Woodward and P. Colella, The Numerical Simulation of Two-Dimensional Fluid Flow with Strong Shocks, J. Comput. Phys, 54(1984):115-173.
[63] T. W. Roberts, The Behavior of Flux Difference Splitting Schemes near Slowly Moving Shock Waves, J. Comput. Phys, 10(1990):141-160.
[64] A. Kurganov and E. Tadmor, Solution of two-dimensional Riemann problem for gas dynamics without Riemann problem solvers.
[65] C. W. Schulz-Rinne, J. P. Collins and H. M. Glaz, Numerical solution of the Riemann problem for two-dimensional gas dynamics, SIAM J. Sci. Comput, 14(1993): 1394-1414.
[66] P. D. Lax and Xu-Dong Liu, Solution of the Riemann problem of two-dimensional gas dynamics by positive schemes, SIAM J. Sci. Comput, 19(1998):319-340.
[67] C. W. Schulz-Rinne, Classification of the Riemann problem for two-dimensional gas dynamics, SIAM J. Math. Anal, 24(1993):76-88.
[68] N.N. Kruzkov, First order quasilinear equations in several independent variables, Math. USSR Sbornik, 10(1970):217-243.
[69] J. P. Vila, High-order schemes and entropy condition for nonlinear hyperbolic systems of conservation laws, Mathematics of computation, 50(1986)53-73.
[70] A.Harten, On the symmetric form of systems of conservation laws with entropy, J. Comput. Phys, 49(1983):151-164.
[71] S. Osher, Riemann solver, the entropy condition, and difference approximations, SIAM J. Num. Anal, 21(1984):217-235.
[72] S. Osher and A. Majda, Numerical viscosity and the entropy condition, Communications on Pure and Applied Mathmatics, (1979):797-838.
[73] S. Osher and S. Chakravarthy, High resolution schemes and the entropy condition, SIAM J.Num.Anal, 21(1984):955-984.
[74] T. Zhang and Y. Zheng, Conjecture on the structure on the solutions of the Riemann problem for two-dimensional gas dynamics systems, SIAM J. Math. Anal, 21(1990):593-630.
[75] T. Chang, G. Q. Chen and S. Yang, On the 2-D Riemann problem for the compressible Euler equations, I. Interaction of shock and rarefaction waves, Discrete Contin. Dynam. Systems, l(1995):555-584.
[76] T. Chang, G. Q. Chen and S. Yang, On the 2-D Riemann problem for the compressible Euler equations, II. Interaction of contact discontinuities, Discrete Contin. Dynam. Systems, 6(2000) :419-430.