一维守恒型方程（组）的熵耗散格式
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摘要
本文考虑的一维双曲守恒律方程(组)。我们给出了一种设计高分辨率格式的方法。此格式是Godunov型的，用的是分片线性重构。与传统的Godunov型格式不同的是此格式在计算过程中不仅计算数值解还计算了数值熵。在每个网格中线性重构函数的斜率是根据熵耗散得到的，它要求此网格上重构函数熵的网格平均应与此网格上的数值熵相等。数值解是根据有限体积法(finite-volume)求得的，同时数值熵计算的时候格式中有一熵耗散项，它在计算过程中耗散熵。通过这种方式为数值计算引入了适当的粘性，稳定了计算。
在[25]中我们对于特殊的熵函数分析了格式的精度。本文中对于一般的熵函数，我们分析了格式的精度，格式在远离极值点附近为二阶精度，在极值点附近为一阶精度。因为格式与传统的守恒型格式不同，所以我们给出了它的相容性定义和Lax-Wendroff定理，定理说如果用我们的数值格式求得的数值解收敛，则它一定收敛到方程的满足熵条件的弱解。
设计这样一种格式的一个重要动机是期望用此类格式来克服守恒型方程组的数值模拟中，线性特征场上的数值耗散问题。为此对于线性传输方程，我们用不带熵耗散函数的格式进行了一些数值实验，以研究数值熵对消除线性耗散问题的作用。在此研究的基础上，我们设计了两种格式，一种格式中我们仍给线性方程或方程组的线性特征场以一定的熵耗散以稳定计算。另一种格式中我们保持线性方程或方程组的线性特征场上的熵守恒，同时为数值解的重构设计了一种所谓的“极大值减少极小值增”(Minimums-Increase-and-Maximums-Decrease或MIMD)斜率控制因子以消除间断附近的非物理振荡。
最后我们分别给出了用带线性熵耗散的格式和不带线性熵耗散但在数值解的重构中运用了MIMD斜率控制因子的格式进行数值实验，包括单个方程的和方程组的。从中我们可看出熵耗散因子是如何抑制非物理的振荡的，以及格式对计算的有效性。
In this paper, we are concerned with hyperbolic conservation laws in one space dimension. We describe a method to design high resolution schemes for the equations. The designed scheme is of Godunov-type with piecewise-line reconstruction. Different from all other Godunov-type schemes for the conservation laws, our scheme computes not only the numerical solution but also an approximation to the entropy, called numerical entropy. In the scheme the reconstruction of solution is performed by requiring the cell-average of the entropy of the reconstructed solution to be equal to the numerical entropy in each grid cell. Both the numerical solution and numerical entropy are computed in a finite-volume fashion while the computation of the latter involves a so-called entropy dissipation term, which simulates the variation of the entropy. In doing so, the numerical dissipation is introduced in the scheme to stabilize the computation.
The definition of consistency of the schemes is given and a Lax-Wendroff theory for the scheme is also given, which says that if the numerical solution converges, it converges to an entropy solution of the orginal equation.
Since a major motivation of designing this kind of scheme is to try to overcome the numerical dissipation in the linearly degenerated characteristic field in the system case, we conduct a numerical study of the scheme without entropy dissipation term on the linear advection equation to investigate the way in which the numerical entropy eliminates the linear dissipation. Based on this study, we designed two schemes of the type. In one of the schemes, certain entropy dissipation term is still involved in the computation of the entropy in the linearly degenerated characteristic field; thus, the numerical entropy is not coservative there. In the other scheme, we set the entropy dissipation in the linear field to be zero so that the numerical entropy is conservative in this field. We then design a so-called "Minimums-Increase-and-Maximums-Decrease" (MIMD) slope-limiter in the reconstruction step of the scheme to eliminate the non-physical oscillation then caused.
在[25]中我们对于特殊的熵函数分析了格式的精度。本文中对于一般的熵函数，我们分析了格式的精度，格式在远离极值点附近为二阶精度，在极值点附近为一阶精度。因为格式与传统的守恒型格式不同，所以我们给出了它的相容性定义和Lax-Wendroff定理，定理说如果用我们的数值格式求得的数值解收敛，则它一定收敛到方程的满足熵条件的弱解。
设计这样一种格式的一个重要动机是期望用此类格式来克服守恒型方程组的数值模拟中，线性特征场上的数值耗散问题。为此对于线性传输方程，我们用不带熵耗散函数的格式进行了一些数值实验，以研究数值熵对消除线性耗散问题的作用。在此研究的基础上，我们设计了两种格式，一种格式中我们仍给线性方程或方程组的线性特征场以一定的熵耗散以稳定计算。另一种格式中我们保持线性方程或方程组的线性特征场上的熵守恒，同时为数值解的重构设计了一种所谓的“极大值减少极小值增”(Minimums-Increase-and-Maximums-Decrease或MIMD)斜率控制因子以消除间断附近的非物理振荡。
最后我们分别给出了用带线性熵耗散的格式和不带线性熵耗散但在数值解的重构中运用了MIMD斜率控制因子的格式进行数值实验，包括单个方程的和方程组的。从中我们可看出熵耗散因子是如何抑制非物理的振荡的，以及格式对计算的有效性。
In this paper, we are concerned with hyperbolic conservation laws in one space dimension. We describe a method to design high resolution schemes for the equations. The designed scheme is of Godunov-type with piecewise-line reconstruction. Different from all other Godunov-type schemes for the conservation laws, our scheme computes not only the numerical solution but also an approximation to the entropy, called numerical entropy. In the scheme the reconstruction of solution is performed by requiring the cell-average of the entropy of the reconstructed solution to be equal to the numerical entropy in each grid cell. Both the numerical solution and numerical entropy are computed in a finite-volume fashion while the computation of the latter involves a so-called entropy dissipation term, which simulates the variation of the entropy. In doing so, the numerical dissipation is introduced in the scheme to stabilize the computation.
The definition of consistency of the schemes is given and a Lax-Wendroff theory for the scheme is also given, which says that if the numerical solution converges, it converges to an entropy solution of the orginal equation.
Since a major motivation of designing this kind of scheme is to try to overcome the numerical dissipation in the linearly degenerated characteristic field in the system case, we conduct a numerical study of the scheme without entropy dissipation term on the linear advection equation to investigate the way in which the numerical entropy eliminates the linear dissipation. Based on this study, we designed two schemes of the type. In one of the schemes, certain entropy dissipation term is still involved in the computation of the entropy in the linearly degenerated characteristic field; thus, the numerical entropy is not coservative there. In the other scheme, we set the entropy dissipation in the linear field to be zero so that the numerical entropy is conservative in this field. We then design a so-called "Minimums-Increase-and-Maximums-Decrease" (MIMD) slope-limiter in the reconstruction step of the scheme to eliminate the non-physical oscillation then caused.
引文
[1] F. Bouchut, An Antidiffusive entropy scheme for monotone scalar conservation laws, J. Sci. Comput., 21 (2004), pp. 1-30.
[2] F. Bouchut, Ch. Bourdarias & B. Perthame, A MUSCL method satisfying all numerical entropy inequalities, Math. Comp., 65 (1996), pp. 1439-1461.
[3] A. J. Chorin and J. E. Marsden, A mathematical introduction to fluid mechanics, Springer verlag, New York, Heidelberg, Berlin, 1993.
[4] P. Colella and P. R. Woodward, The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54(1984), pp. 115-173.
[5] P. Colella and P. R. Woodward, The piecewise-parabolic method(PPM) for gas dynamical simulation, J. Comput. Phys., 54 (1984), pp. 174-201.
[6] Despres, B and Lagoutiere, F, Contact discontinuity capturing schemes for linear advection and compressible gas dynamics, J. Sci. Comput., 16 (2001), pp. 479-524.
[7] K. O. Friedrichs and P. D. Lax, Systems of conservation equations with a convex extension, Proc. Nat. Acad. Sci., 68 (1971), pp. 1686-1688.
[8] S. K. Godunov, The problems of a generalized solution in the theory of quasi-lincar equations and in gas dynamics, Russian Math. Surv., 17(1962), pp. 145-156.
[9] S. K. Godunov, A different method for numerical calculation of discontinuous solution of the equations of hydrodynamics, Mat. Sb., 47(1959), pp. 271-306.
[10] A. Harten, ENO schemes with subcell resolution, J. Comput. Phys., 83(1987), pp. 148-184.
[11] A. Harten, and S. Osher, Uniformly high-order accurate nonoscillatory scheme I~*, SIAM J. Numer. Anal., 24 (1987), pp. 279-309.
[12] A. Harten, B. Engquist, S. Osher, and S. R. Chakravarthy, Unifomly high order accurate essentially non-oscillatory schemes, Ⅲ, J. Comput. Phys., 71 (1987), pp. 231-303.
[13] A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 49(1983), pp. 357-393.
[14] A. Harten, The artifical compression method for computation of shocks and contact discontinuities. 1. Single conservation laws, Comput. Pure Appl. Math., 30 (1977), pp. 611-638.
[15] G. Jiang and C. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys. 126 (1996), pp. 202-228.
[16] P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, SIAM(1972).
[17] P. D. Lax, Shock waves and entropy, in Contributions to Nonlinear Functionial Analysis(E.A. Zarantonello, ed.), Academic Press, New York, 1971, pp. 603-634.
[18] P. D. Lax, Hyperbolic system of conservation laws, Ⅱ, J. Comm. Pure Appl. Math., 10(1957), pp. 537-566.
[19] P. D. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical computation, Commun. Pure. Apply. Math. 7 (1954), pp. 159-193.
[20] R. J. LeVeque, Finite volume methods for hyperbolic problems, Published by the press Syndicate of the University of Cambridge, 2002.
[21] R. J. LeVeque, Numerical methods for conservation laws, Birkhauser-verla, Basel, Boston, Berlin, 1990.
[22] H. Li, and D. Mao Entropy dissipation scheme and MIMD slope limiter, submitted to J. Sci. Comput..
[23] H. Li, and D. Mao Entropy dissipation scheme for hyperbolic systems of conservation laws in one space dimension, submitted to J. Sci. Comput..
[24] H. Li, and D. Mao The design of the entropy dissipator of the entropy diaaipating scheme for scalar conservation law, Chinese J. Comput. Phys., 21(2004), pp. 319-326.
[25] H. Li, Second-order entropy dissipation scheme for scalar conservation laws in one space dimension, Master's thesis, No. 11903-99118086, Shanghai University (in Chinese).
[26] K. Lie, and S. Noelle, On the artificial compression method for second-order nonoscillatory central difference schemes for systems of conservation laws, SIAM J. Sci. Comput., 24 (2003), pp. 1157-1174.
[27] X.D. Liu & E. Tadmor, Third order nonscillatory central schemes for hyperbolic conservation laws, Numer. Math 79 (1998), pp 397-425.
[28] D. Mao, Entropy satisfaction of a conservative shock tracking method, SIAM J. Numer. Anal.,36 (1999), pp. 529-550.
[29] N. Nessyahu & E. Tadmor, Non-oscillatory central difference for hyperbolic conservation laws, J. Comput. Phys., 87 (1990), pp408-448.
[30] A. Kurganov & E. Tadmor, New High-resolution central schemes for non-linear conservation laws and convection diffusion equations, J. Comput. Phys., 160, (2000), pp. 241-282.
[31] P. L. Roe, Characteristic-based schemes for the Euler equations, Ann. Rev. Fluid Mech., 18(1986), pp. 337-365.
[32] P. L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, J.Comput.Phys., 43(1981), pp. 357-372.
[33] Chi-wang Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, Lecture Notes in Math. 1697, Springer, 1998, pp. 325-432.
[34] Chi-wang Shu, Efficient implementation of essentially non-oscillatory shock-capturing schemes, Ⅱ, J. Comput. Phys., 83 (1989), pp. 32-78.
[35] Chi-Wang Shu, TVB uniformly high-order schemes for conservation laws, Math. Comput., 49(1987), pp. 105-121.
[36] S. Osher, and S. Chakravarthy, High resolution schemes and the entropy condition, STAM J. Numer. Anal., 21 (1984), pp. 955-984.
[37] J. Smoller, Shock waves and reaction-diffusion equations, Second Edition, SpringerVerlag, New York, Berlin, Heidelberg, 1999.
[38] G. A. Sod, A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys., 27(1978), pp. 1-31.
[39] P.K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer. Anal. 21,995 (1984).
[40] E. Tadmor, Entropy functions for symmetric systems of conservation laws, J. Math. Anal. Appl., 122 (1987), pp. 355-359.
[41] B. Van Leer, Towards the ultimate conservative scheme Ⅲ. Upstream-centered finitedifference schemes for ideal compressible flow, J. Comput. Phys., 23(1977), pp. 263-275.
[42] B. Van Leer, Towards the ultimate conservative scheme Ⅱ. Monotonicity and conservation combined in a second order scheme, J. Comput. Phys.,14(1974), pp. 361-370.
[43] B. Van Leer,Towards the ultimate conservative scheme I. The quest of Monotonicity, Springer Lecture Notes in Physics. 18(1973), pp. 163-168.
[44] G. B. Whitham, Linear and Nonlinear Waves, New York: Wiley, 1974.
[45] 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)．
[46] H. Yang, An artifical compression method for ENO schemes. The slope modification method, J. Comput. Phys., 89 (1990), pp. 125-160.
[47] 周毓麟，一维非定常流体力学，北京：科学出版社，(1998)．
[48] 应隆安，腾振寰，双曲守恒律方程及其差分方法，北京：科学出版社，(1991)．
[2] F. Bouchut, Ch. Bourdarias & B. Perthame, A MUSCL method satisfying all numerical entropy inequalities, Math. Comp., 65 (1996), pp. 1439-1461.
[3] A. J. Chorin and J. E. Marsden, A mathematical introduction to fluid mechanics, Springer verlag, New York, Heidelberg, Berlin, 1993.
[4] P. Colella and P. R. Woodward, The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54(1984), pp. 115-173.
[5] P. Colella and P. R. Woodward, The piecewise-parabolic method(PPM) for gas dynamical simulation, J. Comput. Phys., 54 (1984), pp. 174-201.
[6] Despres, B and Lagoutiere, F, Contact discontinuity capturing schemes for linear advection and compressible gas dynamics, J. Sci. Comput., 16 (2001), pp. 479-524.
[7] K. O. Friedrichs and P. D. Lax, Systems of conservation equations with a convex extension, Proc. Nat. Acad. Sci., 68 (1971), pp. 1686-1688.
[8] S. K. Godunov, The problems of a generalized solution in the theory of quasi-lincar equations and in gas dynamics, Russian Math. Surv., 17(1962), pp. 145-156.
[9] S. K. Godunov, A different method for numerical calculation of discontinuous solution of the equations of hydrodynamics, Mat. Sb., 47(1959), pp. 271-306.
[10] A. Harten, ENO schemes with subcell resolution, J. Comput. Phys., 83(1987), pp. 148-184.
[11] A. Harten, and S. Osher, Uniformly high-order accurate nonoscillatory scheme I~*, SIAM J. Numer. Anal., 24 (1987), pp. 279-309.
[12] A. Harten, B. Engquist, S. Osher, and S. R. Chakravarthy, Unifomly high order accurate essentially non-oscillatory schemes, Ⅲ, J. Comput. Phys., 71 (1987), pp. 231-303.
[13] A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 49(1983), pp. 357-393.
[14] A. Harten, The artifical compression method for computation of shocks and contact discontinuities. 1. Single conservation laws, Comput. Pure Appl. Math., 30 (1977), pp. 611-638.
[15] G. Jiang and C. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys. 126 (1996), pp. 202-228.
[16] P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, SIAM(1972).
[17] P. D. Lax, Shock waves and entropy, in Contributions to Nonlinear Functionial Analysis(E.A. Zarantonello, ed.), Academic Press, New York, 1971, pp. 603-634.
[18] P. D. Lax, Hyperbolic system of conservation laws, Ⅱ, J. Comm. Pure Appl. Math., 10(1957), pp. 537-566.
[19] P. D. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical computation, Commun. Pure. Apply. Math. 7 (1954), pp. 159-193.
[20] R. J. LeVeque, Finite volume methods for hyperbolic problems, Published by the press Syndicate of the University of Cambridge, 2002.
[21] R. J. LeVeque, Numerical methods for conservation laws, Birkhauser-verla, Basel, Boston, Berlin, 1990.
[22] H. Li, and D. Mao Entropy dissipation scheme and MIMD slope limiter, submitted to J. Sci. Comput..
[23] H. Li, and D. Mao Entropy dissipation scheme for hyperbolic systems of conservation laws in one space dimension, submitted to J. Sci. Comput..
[24] H. Li, and D. Mao The design of the entropy dissipator of the entropy diaaipating scheme for scalar conservation law, Chinese J. Comput. Phys., 21(2004), pp. 319-326.
[25] H. Li, Second-order entropy dissipation scheme for scalar conservation laws in one space dimension, Master's thesis, No. 11903-99118086, Shanghai University (in Chinese).
[26] K. Lie, and S. Noelle, On the artificial compression method for second-order nonoscillatory central difference schemes for systems of conservation laws, SIAM J. Sci. Comput., 24 (2003), pp. 1157-1174.
[27] X.D. Liu & E. Tadmor, Third order nonscillatory central schemes for hyperbolic conservation laws, Numer. Math 79 (1998), pp 397-425.
[28] D. Mao, Entropy satisfaction of a conservative shock tracking method, SIAM J. Numer. Anal.,36 (1999), pp. 529-550.
[29] N. Nessyahu & E. Tadmor, Non-oscillatory central difference for hyperbolic conservation laws, J. Comput. Phys., 87 (1990), pp408-448.
[30] A. Kurganov & E. Tadmor, New High-resolution central schemes for non-linear conservation laws and convection diffusion equations, J. Comput. Phys., 160, (2000), pp. 241-282.
[31] P. L. Roe, Characteristic-based schemes for the Euler equations, Ann. Rev. Fluid Mech., 18(1986), pp. 337-365.
[32] P. L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, J.Comput.Phys., 43(1981), pp. 357-372.
[33] Chi-wang Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, Lecture Notes in Math. 1697, Springer, 1998, pp. 325-432.
[34] Chi-wang Shu, Efficient implementation of essentially non-oscillatory shock-capturing schemes, Ⅱ, J. Comput. Phys., 83 (1989), pp. 32-78.
[35] Chi-Wang Shu, TVB uniformly high-order schemes for conservation laws, Math. Comput., 49(1987), pp. 105-121.
[36] S. Osher, and S. Chakravarthy, High resolution schemes and the entropy condition, STAM J. Numer. Anal., 21 (1984), pp. 955-984.
[37] J. Smoller, Shock waves and reaction-diffusion equations, Second Edition, SpringerVerlag, New York, Berlin, Heidelberg, 1999.
[38] G. A. Sod, A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys., 27(1978), pp. 1-31.
[39] P.K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer. Anal. 21,995 (1984).
[40] E. Tadmor, Entropy functions for symmetric systems of conservation laws, J. Math. Anal. Appl., 122 (1987), pp. 355-359.
[41] B. Van Leer, Towards the ultimate conservative scheme Ⅲ. Upstream-centered finitedifference schemes for ideal compressible flow, J. Comput. Phys., 23(1977), pp. 263-275.
[42] B. Van Leer, Towards the ultimate conservative scheme Ⅱ. Monotonicity and conservation combined in a second order scheme, J. Comput. Phys.,14(1974), pp. 361-370.
[43] B. Van Leer,Towards the ultimate conservative scheme I. The quest of Monotonicity, Springer Lecture Notes in Physics. 18(1973), pp. 163-168.
[44] G. B. Whitham, Linear and Nonlinear Waves, New York: Wiley, 1974.
[45] 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)．
[46] H. Yang, An artifical compression method for ENO schemes. The slope modification method, J. Comput. Phys., 89 (1990), pp. 125-160.
[47] 周毓麟，一维非定常流体力学，北京：科学出版社，(1998)．
[48] 应隆安，腾振寰，双曲守恒律方程及其差分方法，北京：科学出版社，(1991)．