正则长波方程的特征块中心差分方法
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摘要
正则长波方程简称RLw(Regularised Long wave)方程，是流体力学中一个著名的非线性波动方程．在偏微分方程以及许多物理现象中，如孤立子理论，浅水波和离子声波等问题中，正则长波方程占据着十分重要的地位．
首先，简要介绍了本文所研究问题的背景和现状，并简单介绍本文所做的工作及用到的基本理论知识．
其次，针对正则长波方程的初边值问题，分别给出了一维线性和二维线性正则长波方程的特征块中心差分方法．不但得到了方程的近似解，也得到了解的一阶导数的近似值及其误差估计，其近似解按照离散L2。范数达到最优阶误差估计．所讨论方法的近似解和基于二次插值的特征差分法的近似解具有相同阶的误差估计，解得一阶导数的近似值则具有超收敛性，达到了同近似解相同的精度．
最后，结合特征块中心差分方法，给出实际的数值试验算例．结果表明解的一阶导数的近似值同近似解具有相同的精度，数值试验说明了该方法的有效性和可行性．
Regular long wave equation that it is called RLW equation, it is an important nonlinearwave equation in ?uid mechanics. Regularised Long Wave equation plays a major role in alarge number of partial di?erential equations. They can describe many important physicalphenomenons and equations, such as soliton theory, shallow water waves and ion acousticphasma waves and so on.
First, a brief introduction is given to the latest development of these approaches. Thenwe introduce brie?y our work and the basic theory to be applied in this paper.
Second, for the initial boundary value problem of RLW equation. The method ofcharacteristic block center di?erence for one-dimensional and two-dimensional linear reg-ularized long wave equation is given. The approximate solution of equation is obtained,we also get solution,s first order derivative and error estimation. The discrete L2 normmost optimal order number error estimation for approximate solution. The error order ofthe approximate solution is the same as one of the characteristic di?erence method basedon quadratic interpolation, and its first derivatives show super convergent. The first orderderivative approximate solution with approximate solution has the same accuracy.
Finally, according to characteristic block center di?erence scheme of Regular long waveequation. The actual calculation example is given. The results show that the first orderderivative approximate solution with approximate solution has the same accuracy, and alsoshow that the method is e?ective and feasible.
首先，简要介绍了本文所研究问题的背景和现状，并简单介绍本文所做的工作及用到的基本理论知识．
其次，针对正则长波方程的初边值问题，分别给出了一维线性和二维线性正则长波方程的特征块中心差分方法．不但得到了方程的近似解，也得到了解的一阶导数的近似值及其误差估计，其近似解按照离散L2。范数达到最优阶误差估计．所讨论方法的近似解和基于二次插值的特征差分法的近似解具有相同阶的误差估计，解得一阶导数的近似值则具有超收敛性，达到了同近似解相同的精度．
最后，结合特征块中心差分方法，给出实际的数值试验算例．结果表明解的一阶导数的近似值同近似解具有相同的精度，数值试验说明了该方法的有效性和可行性．
Regular long wave equation that it is called RLW equation, it is an important nonlinearwave equation in ?uid mechanics. Regularised Long Wave equation plays a major role in alarge number of partial di?erential equations. They can describe many important physicalphenomenons and equations, such as soliton theory, shallow water waves and ion acousticphasma waves and so on.
First, a brief introduction is given to the latest development of these approaches. Thenwe introduce brie?y our work and the basic theory to be applied in this paper.
Second, for the initial boundary value problem of RLW equation. The method ofcharacteristic block center di?erence for one-dimensional and two-dimensional linear reg-ularized long wave equation is given. The approximate solution of equation is obtained,we also get solution,s first order derivative and error estimation. The discrete L2 normmost optimal order number error estimation for approximate solution. The error order ofthe approximate solution is the same as one of the characteristic di?erence method basedon quadratic interpolation, and its first derivatives show super convergent. The first orderderivative approximate solution with approximate solution has the same accuracy.
Finally, according to characteristic block center di?erence scheme of Regular long waveequation. The actual calculation example is given. The results show that the first orderderivative approximate solution with approximate solution has the same accuracy, and alsoshow that the method is e?ective and feasible.
引文
[1] C Eilbeck，G R Meguire Numerical study of the regularized long wave equa- tion I：Numerical Methods[J]J Comput Phys，1975，19：43—57
[2]L Wahllin Error estimates for a Galerkin method for a class of model equations for long waves[J]Numer Math，1975，23：289—303
[3]B Y Guo，B S Manoranjan A spectral method for solving the RLW equation IMA[J].J Numer Anal，19s5(3)：307—318
[4]朱江．非线性RLW方程的特征数值方法[J],应用数学学报，1990，13(1)：64—73
[5]D H Peregrine Long waves on a beach[J]J Fluid Mech，1967，27：815—827
[6]D Bhardwaj，R A Shankar Computational method for Regularized long wave equation[J] Comput Math Appl，2000，40：1397—1404
[7]L Zhang，Q A Chang New finite difference method for regularized long wave equation[J] Chinese J Numer Method Comput Appl．2001．23：58—66
[8]Hu Yingying，Wang Yushun，Wang Huiping A new explicit multisymplectic scheme for the RLW equation[J]Chinese Math Numer Sin，2009，31(4)：349—362
[9]A A Soliman，H A Abdo New exact solutions of nonlinear variants of the RLW and PHI—four and Boussinesq equations based on modified extended direct algebraic method[J]Int J Nonlinear Sci，2009，7(3)：274—282
[10]Dag idris，Korkmaz Alper，Saka B n lent Cosine expansion—based differential quadra- ture algorithm for numerical solution of the RLW equation[J]Numer Methods Partial Differential Equations，2010，26(3)：544—560
[11]Dag idris，Dereli Yilmaz Numerical solution of RLW equation using radial basis func—tions[J]Int J Comput Math，2010，87(1)：63—76
[12]A H A Ali Chebyshev collocation spectral method for solving the RLW equation[J]Int J Nonlinear Sci，2009，7(2)：131—142
[13]Siraj—ul—Islam，Haq Sirajul，Ali Arshed A mesh free method for the numerical solution of the RLW equation[J]J Comput Appl Math，2009，223(2)：997—1012
[14]Gu Haiming，Chen Ning Least squares mixed finite element methods for the RLW equa- tions[J]Numer Methods Partial Differential Equations，2008，24(3)：749—758
[15]A A Soliman Exact traveling wave solution of nonlinear variants of the RLW and the PHI一％ur equations[J]Phys Lett A，2007，368(5)：383—390
[16]Lou Yijun Bifurcation of travelling wave solutions in a nonlinear variant of the RLW equation[J]Commun Nonlinear Sci Numer Simul，2007，12(8)：1488—1503[州M Rafei，D D Ganji，Mohammadi Application of homotopy perturbation method to the RLW and generalized modified Boussinesq equations[J]Phys Lett A，2007，364(1)：1-6
[18]Wazwaz Abdul Majid Analytic study on nonlinear variants of the RLW and the PHI一％ur equations[J]Commun Nonlinear Sci Numer Simul，2007，12(3)：314—327
[19]J I Ramos Explicit finite difference methods for the EW and RLW equations[J]Appl Math Comput，2006，179(2)：622—638
[20]Tan Junyu A class of analytic solutions for the RLW Burgers equation[J]Chinese Math Practice Theory,2001，31(5)：545—549
[21]A A Soliman，K R Raslan Collocation method using quadratic B—spline for the RLW equa- tion[J]Int J Comput Math，2001，78(3)：399—412
[22]Guo Ben—yu，V S Manoranjian A spectral method for solving the RLW equation[J] IMA Number Anal．1985．5：307—318
[23]张鲁明，常谦顺．正则长波方程的一个新的差分方法[J].数值计算与计算机应用，2000；12(4)：247—254
[24]邬华谟，郭本瑜．Kdv—Burgers—RLW方程的高精度差分格式[J].计算数学，1983，5(1)：90—98
[25]L R T Gardner，G A Gardner，A Dogan A least—sqnares finite elelment scheme for the RLW equation[J]Comm Numer Methods Engrg，1996，12：795—804
[26]L R T Gardner，G A Gardner Solitary waves of the regularized long wave equation[J] Comput Phys，1990，91：441—459
[27]z D Luo，R X Liu Mixed finite element analysis and numerical solitary solution for the RLW equation[J]SIAM J Numer Anal，1998，36(1)：89—104
[28]任宗修，郝岩，任卫银．非线性RLW方程的经济型差分一流线扩散法[J].工程数学学报，2008，25：708—712
[29]J Douglas，T F Russell Numerical nethods for convection—dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures，SIAM J Numer Anal 1982，19：871—884
[30]You Tongshun The characterictic difference m~hod for nonlinear convection—diffusion problems in two space variables[J]Math Numer Sinica，1993，15(4)：402—409
[31]Zhang Zhiyue，Du Ning Characteristics difference and finite element methods for neurve conduction equation[J]应用数学，2001，14(2)：12—14
[32]z Y Zhang The new"ADI charcteristics finite element methods for the three-dimensional nerve conduction equation[J]N P Sci Comp，2002，10：355—370
[33]Wang Tongke Characteristic difference method with moving mesh for one di—mensional nonlinear Convection dominated diffusion equation[J]Chinese J Comput Physi，2003，20(6)：493—497
[34]A Weiser and M F Wheeler On convergence of block—centered finite differences for elliptic problems[J]SIAM J Numer Anal，1988，25：351—375
[35]Wang Shenlin Blocked centered different methods for quasi linear hyperbolic partia integro—differential equations and boundary value problems[R]London World Scientific，199l 994—999
[36]王申林，孙淑英．对流扩散问题的特征块中心差分法[J].计算数学，1999(4)：464—473
[37]王震，谢树森．海水入侵数值模拟的特征块中心差分法[J]应用数学学报，2006(4)：633—645
[38]郭双冰，张志跃．一类非线性发展方程的特征中心差分法[J].计算物理，2007(6)：637—645
[39]应隆安．有限元方法讲义[M]._北京：北京大学出版社，1988
[40]胡健伟，汤怀民．微分方程数值方法[Ml_北京：科学出版社，2003
[41]伍卓群，尹景学，王春朋．椭圆与抛物型方程引论[Ml_吉林：吉林大学出版社，2000
[42]李荣华，冯果忱．微分方程数值解法[Ml_北京：高等教育出版社，2003
[43]C Johnson Numerical solution of partial differential equation by the finite element method[M]London Cambridge University Press，1987
[44]黄正洪．二维RLW方程的Cauchy问题[J]._应用数学与力学，2002，23(2)：157—164
[45]尚亚东，钮鹏程．二维RLW方程和二维SRLW方程的显式精确解[J].应用数学，1998；11(3)：1—5
[46]A Dogan Numerical solution of regularized long wave equation using Petrov—Galerkin method[J]Comm Num Meth Eng，2001，17：485—494
[47]Q Chang，G Wang，B Guo Conservative Scheme for a Model of Nonlinear Dispersive Wave and Its Solitary Waves Induced by Boundary notion[J]J Comput Phys，1991，93：360—375
[48]M Lenior Optimal Isoparametric Finite Elements and Error Estimates for Domains In—volving Curved Boundaries[J]SIAM J Numer Anal，1986，23：562—580
[49]黄明游．发展方程数值计算方法[M].北京：科学出皈社，2004：4—6
[50]D H Peregrine Calculation of the development of an undular vore[J]J Fluid Mech，1966．25：321-330
[51] T.B.Benjamin, J.J.Mahony. Model equations for wave in nonlinear dispersive systems[j].Philos Trans Roy Soc London Ser A, 1972,272: 47-78.
[52] Wu Haijun, Li Yonghai, Li Ronghua. Adaptive generalized di?erence/finite Volume Com-putations for two dimensional nonliear parabolic equation[J]. Chinese J Comput Phys,2003,20(4): 298-306.
[2]L Wahllin Error estimates for a Galerkin method for a class of model equations for long waves[J]Numer Math，1975，23：289—303
[3]B Y Guo，B S Manoranjan A spectral method for solving the RLW equation IMA[J].J Numer Anal，19s5(3)：307—318
[4]朱江．非线性RLW方程的特征数值方法[J],应用数学学报，1990，13(1)：64—73
[5]D H Peregrine Long waves on a beach[J]J Fluid Mech，1967，27：815—827
[6]D Bhardwaj，R A Shankar Computational method for Regularized long wave equation[J] Comput Math Appl，2000，40：1397—1404
[7]L Zhang，Q A Chang New finite difference method for regularized long wave equation[J] Chinese J Numer Method Comput Appl．2001．23：58—66
[8]Hu Yingying，Wang Yushun，Wang Huiping A new explicit multisymplectic scheme for the RLW equation[J]Chinese Math Numer Sin，2009，31(4)：349—362
[9]A A Soliman，H A Abdo New exact solutions of nonlinear variants of the RLW and PHI—four and Boussinesq equations based on modified extended direct algebraic method[J]Int J Nonlinear Sci，2009，7(3)：274—282
[10]Dag idris，Korkmaz Alper，Saka B n lent Cosine expansion—based differential quadra- ture algorithm for numerical solution of the RLW equation[J]Numer Methods Partial Differential Equations，2010，26(3)：544—560
[11]Dag idris，Dereli Yilmaz Numerical solution of RLW equation using radial basis func—tions[J]Int J Comput Math，2010，87(1)：63—76
[12]A H A Ali Chebyshev collocation spectral method for solving the RLW equation[J]Int J Nonlinear Sci，2009，7(2)：131—142
[13]Siraj—ul—Islam，Haq Sirajul，Ali Arshed A mesh free method for the numerical solution of the RLW equation[J]J Comput Appl Math，2009，223(2)：997—1012
[14]Gu Haiming，Chen Ning Least squares mixed finite element methods for the RLW equa- tions[J]Numer Methods Partial Differential Equations，2008，24(3)：749—758
[15]A A Soliman Exact traveling wave solution of nonlinear variants of the RLW and the PHI一％ur equations[J]Phys Lett A，2007，368(5)：383—390
[16]Lou Yijun Bifurcation of travelling wave solutions in a nonlinear variant of the RLW equation[J]Commun Nonlinear Sci Numer Simul，2007，12(8)：1488—1503[州M Rafei，D D Ganji，Mohammadi Application of homotopy perturbation method to the RLW and generalized modified Boussinesq equations[J]Phys Lett A，2007，364(1)：1-6
[18]Wazwaz Abdul Majid Analytic study on nonlinear variants of the RLW and the PHI一％ur equations[J]Commun Nonlinear Sci Numer Simul，2007，12(3)：314—327
[19]J I Ramos Explicit finite difference methods for the EW and RLW equations[J]Appl Math Comput，2006，179(2)：622—638
[20]Tan Junyu A class of analytic solutions for the RLW Burgers equation[J]Chinese Math Practice Theory,2001，31(5)：545—549
[21]A A Soliman，K R Raslan Collocation method using quadratic B—spline for the RLW equa- tion[J]Int J Comput Math，2001，78(3)：399—412
[22]Guo Ben—yu，V S Manoranjian A spectral method for solving the RLW equation[J] IMA Number Anal．1985．5：307—318
[23]张鲁明，常谦顺．正则长波方程的一个新的差分方法[J].数值计算与计算机应用，2000；12(4)：247—254
[24]邬华谟，郭本瑜．Kdv—Burgers—RLW方程的高精度差分格式[J].计算数学，1983，5(1)：90—98
[25]L R T Gardner，G A Gardner，A Dogan A least—sqnares finite elelment scheme for the RLW equation[J]Comm Numer Methods Engrg，1996，12：795—804
[26]L R T Gardner，G A Gardner Solitary waves of the regularized long wave equation[J] Comput Phys，1990，91：441—459
[27]z D Luo，R X Liu Mixed finite element analysis and numerical solitary solution for the RLW equation[J]SIAM J Numer Anal，1998，36(1)：89—104
[28]任宗修，郝岩，任卫银．非线性RLW方程的经济型差分一流线扩散法[J].工程数学学报，2008，25：708—712
[29]J Douglas，T F Russell Numerical nethods for convection—dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures，SIAM J Numer Anal 1982，19：871—884
[30]You Tongshun The characterictic difference m~hod for nonlinear convection—diffusion problems in two space variables[J]Math Numer Sinica，1993，15(4)：402—409
[31]Zhang Zhiyue，Du Ning Characteristics difference and finite element methods for neurve conduction equation[J]应用数学，2001，14(2)：12—14
[32]z Y Zhang The new"ADI charcteristics finite element methods for the three-dimensional nerve conduction equation[J]N P Sci Comp，2002，10：355—370
[33]Wang Tongke Characteristic difference method with moving mesh for one di—mensional nonlinear Convection dominated diffusion equation[J]Chinese J Comput Physi，2003，20(6)：493—497
[34]A Weiser and M F Wheeler On convergence of block—centered finite differences for elliptic problems[J]SIAM J Numer Anal，1988，25：351—375
[35]Wang Shenlin Blocked centered different methods for quasi linear hyperbolic partia integro—differential equations and boundary value problems[R]London World Scientific，199l 994—999
[36]王申林，孙淑英．对流扩散问题的特征块中心差分法[J].计算数学，1999(4)：464—473
[37]王震，谢树森．海水入侵数值模拟的特征块中心差分法[J]应用数学学报，2006(4)：633—645
[38]郭双冰，张志跃．一类非线性发展方程的特征中心差分法[J].计算物理，2007(6)：637—645
[39]应隆安．有限元方法讲义[M]._北京：北京大学出版社，1988
[40]胡健伟，汤怀民．微分方程数值方法[Ml_北京：科学出版社，2003
[41]伍卓群，尹景学，王春朋．椭圆与抛物型方程引论[Ml_吉林：吉林大学出版社，2000
[42]李荣华，冯果忱．微分方程数值解法[Ml_北京：高等教育出版社，2003
[43]C Johnson Numerical solution of partial differential equation by the finite element method[M]London Cambridge University Press，1987
[44]黄正洪．二维RLW方程的Cauchy问题[J]._应用数学与力学，2002，23(2)：157—164
[45]尚亚东，钮鹏程．二维RLW方程和二维SRLW方程的显式精确解[J].应用数学，1998；11(3)：1—5
[46]A Dogan Numerical solution of regularized long wave equation using Petrov—Galerkin method[J]Comm Num Meth Eng，2001，17：485—494
[47]Q Chang，G Wang，B Guo Conservative Scheme for a Model of Nonlinear Dispersive Wave and Its Solitary Waves Induced by Boundary notion[J]J Comput Phys，1991，93：360—375
[48]M Lenior Optimal Isoparametric Finite Elements and Error Estimates for Domains In—volving Curved Boundaries[J]SIAM J Numer Anal，1986，23：562—580
[49]黄明游．发展方程数值计算方法[M].北京：科学出皈社，2004：4—6
[50]D H Peregrine Calculation of the development of an undular vore[J]J Fluid Mech，1966．25：321-330
[51] T.B.Benjamin, J.J.Mahony. Model equations for wave in nonlinear dispersive systems[j].Philos Trans Roy Soc London Ser A, 1972,272: 47-78.
[52] Wu Haijun, Li Yonghai, Li Ronghua. Adaptive generalized di?erence/finite Volume Com-putations for two dimensional nonliear parabolic equation[J]. Chinese J Comput Phys,2003,20(4): 298-306.