线性传输方程和KdV方程满足两个守恒律的差分格式
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摘要
近几年来,茅德康等对线性传输方程设计了一种能保持两个和三个离散守恒律的差分格式(见[44],[45],[15],[48]和[49]),其数值效果无论在解的精度还是长时间的数值模拟方面都远胜于传统的差分格式。
本文的第一个工作是对线性传输方程的保持两个守恒律的差分格式进行了数值分析,揭示了这种格式在计算中各步的误差会相互抵消这一性质。这种性质在目前我们所见过的数值方法中是罕见的。正是因为格式的这种性质,它能远胜于传统的格式。
本文的第二个工作是将这种设计满足线性传输方程多个守恒律的差分方法应用于KdV方程,对之设计了满足两个守恒律的差分格式。具体的做法是将KdV方程分裂成守恒方程部分和散射方程部分,而我们的方法是应用在守恒方程部分的离散上。所设计的格式在长时间的数值模拟中表现出十分优秀的品质。
In recent years, Mao and his co-workers developed difference schemes for linear advection equation which satisfied two or three discrete conservation laws, see [44], [45], [15], [48] and [49]. The numerical results of the developed schemes were far better than traditional difference schemes' at both solutions' accuracy and long-time numerical integrations.
The first work in this paper is to do the numerical analysis for the difference scheme staisfing two discrete conservation laws for the linear advection equation. We reveal that numerical errors of the scheme at successive time steps cancel with each other, which is a feature that is rarely seen in the numerical methods we have ever known. It is this feature of error-self-canceling that makes our scheme far better than traditional schemes.
The second work in this paper is to apply the numerical approach developed for the linear advection equation to the KdV equation. We develop a scheme for the KdV equation, which satisfies the first two conservation laws of the equation. In constructing the scheme, we adopt the splitting strategy to split the equation into the conservation part and the dispersion part, and the approach of designing scheme satisfying two conservation laws is applied in the discretization of the conservation part. The developed scheme shows good quality in long-time numerical simulations.
本文的第一个工作是对线性传输方程的保持两个守恒律的差分格式进行了数值分析,揭示了这种格式在计算中各步的误差会相互抵消这一性质。这种性质在目前我们所见过的数值方法中是罕见的。正是因为格式的这种性质,它能远胜于传统的格式。
本文的第二个工作是将这种设计满足线性传输方程多个守恒律的差分方法应用于KdV方程,对之设计了满足两个守恒律的差分格式。具体的做法是将KdV方程分裂成守恒方程部分和散射方程部分,而我们的方法是应用在守恒方程部分的离散上。所设计的格式在长时间的数值模拟中表现出十分优秀的品质。
In recent years, Mao and his co-workers developed difference schemes for linear advection equation which satisfied two or three discrete conservation laws, see [44], [45], [15], [48] and [49]. The numerical results of the developed schemes were far better than traditional difference schemes' at both solutions' accuracy and long-time numerical integrations.
The first work in this paper is to do the numerical analysis for the difference scheme staisfing two discrete conservation laws for the linear advection equation. We reveal that numerical errors of the scheme at successive time steps cancel with each other, which is a feature that is rarely seen in the numerical methods we have ever known. It is this feature of error-self-canceling that makes our scheme far better than traditional schemes.
The second work in this paper is to apply the numerical approach developed for the linear advection equation to the KdV equation. We develop a scheme for the KdV equation, which satisfies the first two conservation laws of the equation. In constructing the scheme, we adopt the splitting strategy to split the equation into the conservation part and the dispersion part, and the approach of designing scheme satisfying two conservation laws is applied in the discretization of the conservation part. The developed scheme shows good quality in long-time numerical simulations.
引文
[1]U.Ascher and R.McLachlan,Multisymplectic box schemes and the Korteweg-de Vries equation,Appl.Numer.Math.,48,(2004),pp.255-269.
[2]U.Ascher and R.McLachlan,On symplectic and multisymplectic schemes for the KdV equation,J.Sci.Comput.,25,(2005),pp.83-104.
[3]T.Bridges and S.Reich,Numerical methods for Hamiltonia PDEs,J.Phys.A:Math.Gen 39(2006),pp.5287-5320.
[4]A.J.Chorin and J.E.Marsden,A mathematical introduction to fluid mechanics,Springer verlag,New York,Heidelberg,Berlin,1993.
[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]P.Colella,Multidimensional Upwind Methods for Hyperbolic Coservation Laws,J.Comput.Phys.,87(1990),pp.171-200.
[7]Y.Cui and D.Mao,Numerical method satisfying the first two conservation laws for the Korteweg-de Vries equation,J.Comput.Phys.,2007,pp.376-399.
[8]A.Harten,High resolution schemes for hyperbolic conservation laws,J.Comput.Phys.,49(1983),pp.357-393.
[9]A.Harten,S.Osher,B.Engquist,and S.Chakravarthy,Some results on uniformly high-order accurate essentially nonoscillatory schemes,Appl.Nummcr.Math.,2(1986),pp.347-377.
[10]A.Harten,and S.Osher,Uniformly high-order accurate nonascillatory scheme I~*,SIAM J.Numer.Anal.,24(1987),pp.279-309.
[11]A.Harten,B.Engquist,S.Osher and S.R.Chakravarthy,Uniformly high order accurate essentially non-oscillatory schemes,Ⅲ,J.Comput.Phys.,71(1987),pp.231-303.
[12]A.Harten,ENO schemes with subcell resolution,J.Comput.Phys.,83(1989),pp.148-184.
[13]H.Holden,K.Hvistendahl and N.Risebro Operator splitting methods for generalized Korteweg-Dr Vries equations,J.Comput.Phys.,153,(1999),pp.203-222.
[14]P.D.Lax,Shock waves and entropy,in Contributions to Nonlincar Functionial Analysis(E.A.Zarantonello,ed.),Academic Press,New York,1971,pp.603-634.
[15]H.Li,Z.Wang and D.Mao,Numerically neither dissipative nor compressive scheme for linear advection equation and its application to the Euler system,to appcar in J.Sci.Comput.
[16]G.Jiang and C.Shu,Efficieut Implementation of Weighted ENO Schemes,J.Comput.Phys.pp.202-228(1996).
[17]P.D.Lax and B.Wendroff,Systems of conservation laws,Comm.Pure Appl.Math.,13(1960),pp.217-237.
[18]R.J.LeVeque,Finite volume methods for hyperbolic problems,Published by the press Syndicate of the University of Cambridge,2002.
[19]R.J.LeVeque,Numerical methods for conservation laws,Birkhauscr-vcrla,Basel,Boston,Berlin,1990.
[20]K.W.Morton and D.F.Mayers,Mumerieal solution of partial differential equations,Published by the press Syndicate of the University of Cambridge,2005.
[21]F.Nouri and D.Sloan,A comparison of Fourier Pseudospectral methods for the solution of the Korteweg-de Vries Equation,J.Comput.Phys.,83,(1989),pp 324-344.
[22]R.D.Richtmyer and K.W.Morton,Difference Methods for Initial Value Problems,2nd edn.New York,Wiley-Interscience.Reprinted(1994),New York,Kreiger.
[23]P.L.Roe,Some contribution to the modelling of discontinuous flows,Lccturcs in Appl.Math.22,(1985),pp.163-193.
[24]P.L.Roe,Characteristic-based schemes for the Euler equations,Ann.Rev.Fluid Mech.,18(1986),pp.337-365.
[25]P.L.Roe,Approximate Riemann solvers,parameter vectors,and difference schemes,J.Comput.Phys.,43(1981),pp.357-372.
[26]C.Shu,Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws,Lecture Notes in Math.1697,Springcr,1998,pp.325-432.
[27]C.Shu,TVB uniformly high-order schemes for conservation laws,Math.Comput.,49(1987),pp.105-121.
[28]C.Shu and S.Osher,Efficient implementation of essentially non-oscillatory shock capturing schemes,J.Comput.Phys.,77(1988),pp.439-471.
[29]G.Strang,On the construction and comparison of difference schemes,SIAM.J.Numer.Anal.,5,(1968),pp.506-517.
[30]R.Takahashi and T.Ohkawa,Numerical experiment on interaction of solitons describing recurrence of initial data,Computational Mechanics.,5,(1989),pp.273-281.
[31]E.Tadmor,Entropy stability theorey for difference approximations of nonlinear conservation laws and related time-dependent problems,Acta Numcrica(2003),pp.451-512.
[32]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.
[33]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.
[34]B.Van Leer,Towards the ultimate conservative scheme I.The quest of Monotonicity,Springer Lecture Notes in Physics.18(1973),pp.163-168.
[35]Y.Wang,B.Wang and X.Chen,Multisymplectic Euler box scheme for the KdV equation,Chin.Phys.Lett.,24(2007) pp.312-314.
[36]Y.Xu and C.Shu,Local discontinuous Galerkin methods for three classes of nonlinear wave equations,J.Comput.Math.,22,(2004),pp.250-274.
[37]J.Yan and C.Shu,A local discontinuous Galerkin method for KdV type equations,SIAM J.Numer.Anal,40,(2002),pp.769-791.
[38]Zabusky and Kruskal,Interactions of "solitons" in a collisionless plasma and the recurrence of initial states,Phys.Rev.Lett.15(1965),pp.240-243.
[39]P.Zhao and M.Qin,Multisymplectic geometry and multisymplectic Preissmann scheme for the KdV equation,J.Phys.A:Math.Gen 33(2006),pp.3613-3626.
[40]崔艳芬,茅德康,一个解KDV方程的满足两个守恒律的差分格式,应用数学与计算数学学报,2005,19(2):15-22.
[41]邓镇国,马和平,一类非局部非线性色散波方程的Fourier谱方法,上海大学博士论文,11903-04810040.
[42]冯康,秦孟兆,哈密尔顿系统的辛几何算法,浙江科学技术出版社,2003.
[43]郭本瑜,偏微分方程的差分方法,科学出版社,1988.
[44]李红霞,茅德康,单个守恒型方程的熵耗散格式中耗散函数的构造,计算物理,21,No.3(2004),pp.319-331.
[45]李红霞,一维守恒型方程(组)的熵耗散格式,上海大学博士学位论文,No.11903-02820022.
[46]李立康,於崇华,朱政华,微分方程数值解法,复旦大学出版社,上海,(1999).
[47]匡继昌,常用不等式,山东科学技术出版社,济南,(2003).
[48]王志刚,线性传输方程满足多个守恒律的差分格式,上海大学硕士论文,No.11903-03720683.
[49]王志刚,茅德康,线性传输方程满足三个守恒律的差分格式,上海大学学报(自然科学版).
[50]刘儒勋,舒其望,计算流体力学的若干新方法,科学出版社,北京,(2003).
[51]周毓麟,一维非定常流体力学,科学出版社,北京,(1998).
[2]U.Ascher and R.McLachlan,On symplectic and multisymplectic schemes for the KdV equation,J.Sci.Comput.,25,(2005),pp.83-104.
[3]T.Bridges and S.Reich,Numerical methods for Hamiltonia PDEs,J.Phys.A:Math.Gen 39(2006),pp.5287-5320.
[4]A.J.Chorin and J.E.Marsden,A mathematical introduction to fluid mechanics,Springer verlag,New York,Heidelberg,Berlin,1993.
[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]P.Colella,Multidimensional Upwind Methods for Hyperbolic Coservation Laws,J.Comput.Phys.,87(1990),pp.171-200.
[7]Y.Cui and D.Mao,Numerical method satisfying the first two conservation laws for the Korteweg-de Vries equation,J.Comput.Phys.,2007,pp.376-399.
[8]A.Harten,High resolution schemes for hyperbolic conservation laws,J.Comput.Phys.,49(1983),pp.357-393.
[9]A.Harten,S.Osher,B.Engquist,and S.Chakravarthy,Some results on uniformly high-order accurate essentially nonoscillatory schemes,Appl.Nummcr.Math.,2(1986),pp.347-377.
[10]A.Harten,and S.Osher,Uniformly high-order accurate nonascillatory scheme I~*,SIAM J.Numer.Anal.,24(1987),pp.279-309.
[11]A.Harten,B.Engquist,S.Osher and S.R.Chakravarthy,Uniformly high order accurate essentially non-oscillatory schemes,Ⅲ,J.Comput.Phys.,71(1987),pp.231-303.
[12]A.Harten,ENO schemes with subcell resolution,J.Comput.Phys.,83(1989),pp.148-184.
[13]H.Holden,K.Hvistendahl and N.Risebro Operator splitting methods for generalized Korteweg-Dr Vries equations,J.Comput.Phys.,153,(1999),pp.203-222.
[14]P.D.Lax,Shock waves and entropy,in Contributions to Nonlincar Functionial Analysis(E.A.Zarantonello,ed.),Academic Press,New York,1971,pp.603-634.
[15]H.Li,Z.Wang and D.Mao,Numerically neither dissipative nor compressive scheme for linear advection equation and its application to the Euler system,to appcar in J.Sci.Comput.
[16]G.Jiang and C.Shu,Efficieut Implementation of Weighted ENO Schemes,J.Comput.Phys.pp.202-228(1996).
[17]P.D.Lax and B.Wendroff,Systems of conservation laws,Comm.Pure Appl.Math.,13(1960),pp.217-237.
[18]R.J.LeVeque,Finite volume methods for hyperbolic problems,Published by the press Syndicate of the University of Cambridge,2002.
[19]R.J.LeVeque,Numerical methods for conservation laws,Birkhauscr-vcrla,Basel,Boston,Berlin,1990.
[20]K.W.Morton and D.F.Mayers,Mumerieal solution of partial differential equations,Published by the press Syndicate of the University of Cambridge,2005.
[21]F.Nouri and D.Sloan,A comparison of Fourier Pseudospectral methods for the solution of the Korteweg-de Vries Equation,J.Comput.Phys.,83,(1989),pp 324-344.
[22]R.D.Richtmyer and K.W.Morton,Difference Methods for Initial Value Problems,2nd edn.New York,Wiley-Interscience.Reprinted(1994),New York,Kreiger.
[23]P.L.Roe,Some contribution to the modelling of discontinuous flows,Lccturcs in Appl.Math.22,(1985),pp.163-193.
[24]P.L.Roe,Characteristic-based schemes for the Euler equations,Ann.Rev.Fluid Mech.,18(1986),pp.337-365.
[25]P.L.Roe,Approximate Riemann solvers,parameter vectors,and difference schemes,J.Comput.Phys.,43(1981),pp.357-372.
[26]C.Shu,Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws,Lecture Notes in Math.1697,Springcr,1998,pp.325-432.
[27]C.Shu,TVB uniformly high-order schemes for conservation laws,Math.Comput.,49(1987),pp.105-121.
[28]C.Shu and S.Osher,Efficient implementation of essentially non-oscillatory shock capturing schemes,J.Comput.Phys.,77(1988),pp.439-471.
[29]G.Strang,On the construction and comparison of difference schemes,SIAM.J.Numer.Anal.,5,(1968),pp.506-517.
[30]R.Takahashi and T.Ohkawa,Numerical experiment on interaction of solitons describing recurrence of initial data,Computational Mechanics.,5,(1989),pp.273-281.
[31]E.Tadmor,Entropy stability theorey for difference approximations of nonlinear conservation laws and related time-dependent problems,Acta Numcrica(2003),pp.451-512.
[32]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.
[33]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.
[34]B.Van Leer,Towards the ultimate conservative scheme I.The quest of Monotonicity,Springer Lecture Notes in Physics.18(1973),pp.163-168.
[35]Y.Wang,B.Wang and X.Chen,Multisymplectic Euler box scheme for the KdV equation,Chin.Phys.Lett.,24(2007) pp.312-314.
[36]Y.Xu and C.Shu,Local discontinuous Galerkin methods for three classes of nonlinear wave equations,J.Comput.Math.,22,(2004),pp.250-274.
[37]J.Yan and C.Shu,A local discontinuous Galerkin method for KdV type equations,SIAM J.Numer.Anal,40,(2002),pp.769-791.
[38]Zabusky and Kruskal,Interactions of "solitons" in a collisionless plasma and the recurrence of initial states,Phys.Rev.Lett.15(1965),pp.240-243.
[39]P.Zhao and M.Qin,Multisymplectic geometry and multisymplectic Preissmann scheme for the KdV equation,J.Phys.A:Math.Gen 33(2006),pp.3613-3626.
[40]崔艳芬,茅德康,一个解KDV方程的满足两个守恒律的差分格式,应用数学与计算数学学报,2005,19(2):15-22.
[41]邓镇国,马和平,一类非局部非线性色散波方程的Fourier谱方法,上海大学博士论文,11903-04810040.
[42]冯康,秦孟兆,哈密尔顿系统的辛几何算法,浙江科学技术出版社,2003.
[43]郭本瑜,偏微分方程的差分方法,科学出版社,1988.
[44]李红霞,茅德康,单个守恒型方程的熵耗散格式中耗散函数的构造,计算物理,21,No.3(2004),pp.319-331.
[45]李红霞,一维守恒型方程(组)的熵耗散格式,上海大学博士学位论文,No.11903-02820022.
[46]李立康,於崇华,朱政华,微分方程数值解法,复旦大学出版社,上海,(1999).
[47]匡继昌,常用不等式,山东科学技术出版社,济南,(2003).
[48]王志刚,线性传输方程满足多个守恒律的差分格式,上海大学硕士论文,No.11903-03720683.
[49]王志刚,茅德康,线性传输方程满足三个守恒律的差分格式,上海大学学报(自然科学版).
[50]刘儒勋,舒其望,计算流体力学的若干新方法,科学出版社,北京,(2003).
[51]周毓麟,一维非定常流体力学,科学出版社,北京,(1998).