一类非局部非线性色散波方程的Fourier谱方法
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摘要
非局部非线性色散波方程是描述密度分层流体内重力波传播过程的一类模型方程.既然大多数重力内波产生于海水和大气,那么研究这类方程解的性质对于深海石油钻探、水下导航、数值天气预报等都有重要的应用价值,同时对于流体力学、大气学和海洋学等具有重要的理论意义.但是这类方程的色散关系是非局部的,分析方程解的性质不是件容易的事情.因此寻求有效的数值算法是必要的.目前,Fourier变换与非局部算子之间的特殊关系使得Fourier谱或拟谱方法已经成为数值求解这类方程的重要工具.
本文主要解决了三个问题:首先,改进了Pelloni和Dougalis对一类非局部非线性色散波方程(包括Benjamin-Ono方程和中等长波方程)的Fourier谱方法给出的L~2误差估计;其次,最近Thomee和Murthy,Pelloni和Dougalis分别在文章中指出,尽管Fourier拟谱方法对Benjamin-Ono方程在数值计算方面是十分有效的,但没有任何的误差分析,本文的工作很好地回答了这个问题;最后,改进了Maday和Quarteroni对Korteweg-de Vries方程的Fourier谱方法给出的L~2误差估计.
在第三章,对一类非局部非线性色散波方程的周期边界问题建立了能够显式计算的全离散Fourier谱方法逼近格式,对方程的非线性项显式处理和对线性项隐式处理,改进了Pelloni和Dougalis的L~2误差估计,使之提高到丰满(最优),并且能够放宽对时间步长的限制.
在第四章,对最近Thomee和Murthy,Pelloni和Dougalis分别在文章中指出的问题,我们直接对一类非局部非线性色散波方程(包括Benjamin-Ono方程和中等长波方程)建立了能够显式计算的全离散Fourier拟谱逼近格式,利用分数次Sobolev范数度量误差,证明了该格式的稳定性和收敛性,并且具有谱精度.此外,通过一些数值例子表明了本文算法的高精度性和稳定性,并且与其他方法作了比较.
在第五章,将第三章中所涉及的方法和证明技巧成功地推广到Korteweg-deVries方程的周期边界问题,改进了Maday和Quarteroni给出的L~2误差估计,使之提高到丰满(最优).此外,通过数值模拟最近受到关注的初始状态重现实验(zabusky和Kruskal),表明本文算法具有很好的计算稳定性.
在第六章,讨论了一类非局部非线性色散波方程的修正Fourier拟谱逼近格式,证明了该格式的稳定性和收敛性,并且L~2误差估计是丰满(最优)的.
The nonlocal,nonlinear dispersive wave equations are a class of model equations for describing the propagation of gravitational waves in the density stratified fluid.Since most of the gravitational waves arise in the seawater and the atmosphere,studying the properties of the solution to the equations not only has the important application in deepsea oil drilling,underwater navigation and numerical weather prediction,but also has the theoretical significance for fluid mechanics,atmospheric sciences and oceanic sciences. But the dispersive relations of the equations are nonlocal and thus it is not an easy thing to analyze the properties of the solution to the equations.Hence it is necessary to find an efficient numerical method.At present,the Fourier spectral/pseudo-spectral method provides a powerful technique for the numerical solutions of such problems due to the special relations between Fourier transform and nonlocal operator.
This paper mainly solves three problems.First,we improve the error estimates in L~2- norm of the Fourier spectral method for a class of nonlocal,nonlinear dispersive wave equations including the Benjamin-Ono equation and intermediate long wave equation by Pelloni and Dougalis;Second,recently Thomee and Murthy,Pelloni and Dougalis,point out in their papers respectively that Fourier pseudo-spectral method solves the Benjamin-Ono cquaiton well but no error analyses are given,and our works answer the problem well; Third,we improve the error estimates in L~2- norm of the Fourier spectral method for the Korteweg-de Vries equation.
In Chapter 3,we establish the fully discrete spectral method for the explicitly numerical solution to periodic boundary-value problem for two nonlocal,nonlinear dispersive wave equations,the Benjamin-Ono and the Intermediate Long Wave equations.We treat the linear terms in the equation implicitly and the nonlinear terms cxplicitly.We improve the error estimates in L~2-norm by Pelloni and Dougalis and make them optimal.In addition,we relax the restriction on the time-step.
In Chapter 4,for the recent problem pointed out in the papers by Thomnee and Murthy,Pelloni and Dougalis,we directly present the fully discrete spectral method for the explicitly numerical solution to periodic boundary-value problem for two nonlocal, nonlinear dispersive wave equations,the Benjamin-Ono and the Intermediate Long Wave equations.Using the fractional order Sobolev norm for measuring the error,we prove the stability and spectral accuracy of the method.In addition,some numerical examples are given to show the high order and stability of our method and our method is compared with other methods.
In Chapter 5,we successfully generalize the methods and proving skills involved in Chapter 3 to the boundary-value problem for the Korteweg-de Vrics equation.We improve the error estimates in L~2- norm by Maday and Quarteroni and make them optimal.In addition,numerically modeling the recent attractive experiment for the recurrence of initial states by Zabusky and Kruskal,shows that our method has good computational stability.
In Chapter 6,we discuss the modified Fourier pseudo-spectral method for a class of nonlocal,nonlinear dispersive wave equations.We prove the stability and convergence of the method and obtain the optimal error estimates in L~2-norm.
本文主要解决了三个问题:首先,改进了Pelloni和Dougalis对一类非局部非线性色散波方程(包括Benjamin-Ono方程和中等长波方程)的Fourier谱方法给出的L~2误差估计;其次,最近Thomee和Murthy,Pelloni和Dougalis分别在文章中指出,尽管Fourier拟谱方法对Benjamin-Ono方程在数值计算方面是十分有效的,但没有任何的误差分析,本文的工作很好地回答了这个问题;最后,改进了Maday和Quarteroni对Korteweg-de Vries方程的Fourier谱方法给出的L~2误差估计.
在第三章,对一类非局部非线性色散波方程的周期边界问题建立了能够显式计算的全离散Fourier谱方法逼近格式,对方程的非线性项显式处理和对线性项隐式处理,改进了Pelloni和Dougalis的L~2误差估计,使之提高到丰满(最优),并且能够放宽对时间步长的限制.
在第四章,对最近Thomee和Murthy,Pelloni和Dougalis分别在文章中指出的问题,我们直接对一类非局部非线性色散波方程(包括Benjamin-Ono方程和中等长波方程)建立了能够显式计算的全离散Fourier拟谱逼近格式,利用分数次Sobolev范数度量误差,证明了该格式的稳定性和收敛性,并且具有谱精度.此外,通过一些数值例子表明了本文算法的高精度性和稳定性,并且与其他方法作了比较.
在第五章,将第三章中所涉及的方法和证明技巧成功地推广到Korteweg-deVries方程的周期边界问题,改进了Maday和Quarteroni给出的L~2误差估计,使之提高到丰满(最优).此外,通过数值模拟最近受到关注的初始状态重现实验(zabusky和Kruskal),表明本文算法具有很好的计算稳定性.
在第六章,讨论了一类非局部非线性色散波方程的修正Fourier拟谱逼近格式,证明了该格式的稳定性和收敛性,并且L~2误差估计是丰满(最优)的.
The nonlocal,nonlinear dispersive wave equations are a class of model equations for describing the propagation of gravitational waves in the density stratified fluid.Since most of the gravitational waves arise in the seawater and the atmosphere,studying the properties of the solution to the equations not only has the important application in deepsea oil drilling,underwater navigation and numerical weather prediction,but also has the theoretical significance for fluid mechanics,atmospheric sciences and oceanic sciences. But the dispersive relations of the equations are nonlocal and thus it is not an easy thing to analyze the properties of the solution to the equations.Hence it is necessary to find an efficient numerical method.At present,the Fourier spectral/pseudo-spectral method provides a powerful technique for the numerical solutions of such problems due to the special relations between Fourier transform and nonlocal operator.
This paper mainly solves three problems.First,we improve the error estimates in L~2- norm of the Fourier spectral method for a class of nonlocal,nonlinear dispersive wave equations including the Benjamin-Ono equation and intermediate long wave equation by Pelloni and Dougalis;Second,recently Thomee and Murthy,Pelloni and Dougalis,point out in their papers respectively that Fourier pseudo-spectral method solves the Benjamin-Ono cquaiton well but no error analyses are given,and our works answer the problem well; Third,we improve the error estimates in L~2- norm of the Fourier spectral method for the Korteweg-de Vries equation.
In Chapter 3,we establish the fully discrete spectral method for the explicitly numerical solution to periodic boundary-value problem for two nonlocal,nonlinear dispersive wave equations,the Benjamin-Ono and the Intermediate Long Wave equations.We treat the linear terms in the equation implicitly and the nonlinear terms cxplicitly.We improve the error estimates in L~2-norm by Pelloni and Dougalis and make them optimal.In addition,we relax the restriction on the time-step.
In Chapter 4,for the recent problem pointed out in the papers by Thomnee and Murthy,Pelloni and Dougalis,we directly present the fully discrete spectral method for the explicitly numerical solution to periodic boundary-value problem for two nonlocal, nonlinear dispersive wave equations,the Benjamin-Ono and the Intermediate Long Wave equations.Using the fractional order Sobolev norm for measuring the error,we prove the stability and spectral accuracy of the method.In addition,some numerical examples are given to show the high order and stability of our method and our method is compared with other methods.
In Chapter 5,we successfully generalize the methods and proving skills involved in Chapter 3 to the boundary-value problem for the Korteweg-de Vrics equation.We improve the error estimates in L~2- norm by Maday and Quarteroni and make them optimal.In addition,numerically modeling the recent attractive experiment for the recurrence of initial states by Zabusky and Kruskal,shows that our method has good computational stability.
In Chapter 6,we discuss the modified Fourier pseudo-spectral method for a class of nonlocal,nonlinear dispersive wave equations.We prove the stability and convergence of the method and obtain the optimal error estimates in L~2-norm.
引文
[1]T.B.Benjamin.Internal waves of permanent form in fluids of great depth.J.Fluid.Mech.,29:559-592,1967.
[2]Hiroaki Ono.Algebraic solitary waves in stratified fluids.J.Phys.Soc.Japan,39(4):1082-1091,1975.
[3]Rafael Jose Iorio,Jr.On the Cauchy problem for the Benjamin-Ono equation.Comm.Partial Differential Equations,11(10):1031-1081,1986.
[4]K.M.Case.Properties of the Benjamin-Ono equation.J.Math.Phys.,20(5):972-977,1979.
[5]L.Abdelouhab,J.L.Bona,M.Felland,and J.-C.Saut.Nonlocal models for nonlinear,dispersive waves.Phys.D,40:360-392,1989.
[6]M.J.Ablowitz and P.A.Clarkson.Salitons,nonlinear evolution equations and inverse scattering,volume 149 of London Mathematical Society Lecture Note Series.Cambridge University Press,Cambridge,1991.
[7]Elias M.Stein.Harmonic analysis:real-variable methods,orthogonality,and oscillatory integrals,volume 43 of Princeton Mathematical Series.Princeton University Press,Princeton,NJ,1993.With the assistance of Timothy S.Murphy,Monographs in Harmonic Analysis,Ⅲ.
[8]R.I.Joseph.Solitary waves in a finite depth fluid.J.Phys.A,10(12):225-227,1977.
[9]T.Kubota and D.R.S.Ko.Weakly nonlinear,long internal gravity waves in stratified fluids of finite depth.AIAA J.Hydronautics,12:157-165,1978.
[10]M.J.Ablowitz,A.S.Fokas,J.Satsuma,and H.Segur.On the periodic intermediate long wave equation.J.Phys.A:Math.Gen.,15:781-786,1982.
[11]P.M.Santini,M.J.Ablowitz,and A.S.Fokas.On the limit from the intermediate long wave equation to the Benjamin-Ono equation.J.Math.Phys.,25(4):892-899,1984.
[12]C.S.Gardner and G.K.Morikawa.Similarity in the asymptotic behaviour of collision-free hydromagnetic waves and water waves.Technical Report Technical Report NYO - 9082,Courant Institute of Mathematical Sciences,New York University,New York,1960.
[13]A.Jeffrey and T.Kakutani.Weak nonlinear dispersive waves:A discussion centered around the Korteweg-de Vries equation.SIAM Rev.,14:582-643,1972.
[14]Robert M.Miura.The Korteweg-de Vries equation:a survey of results.SIAM Rev.,18(3):412-459,1976.
[15]Mark J.Ablowitz and Harvey Segur.Solitons and the inverse scattering transform,volume 4 of SIAM Studies in Applied Mathematics.Society for Industrial and Applied Mathematics (SIAM),Philadelphia,Pa.,1981.
[16]Alan C.Newell.Solitons in mathematics and physics,volume 48 of CBMS-NSF Regional Conference Series in Applied Mathematics.Society for Industrial and Applied Mathematics (SIAM),Philadelphia,PA,1985.
[17]James W.Cooley and John W.Tukey.An algorithm for the machine calculation of complex Fourier series.Math.Comp.,19:297-301,1965.
[18]Steven A.Orszag.Numerical simulation of incompressible flows within simple boundaries.I.Galerkin(spectral) representations.Studies in Appl.Math.,50:293-327,1971.
[19]David Gottlieb and Steven A.Orszag.Numerical analysis of spectral methods:theory and applications.Society for Industrial and Applied Mathematics,Philadelphia,Pa.,1977.CBMS-NSF Regional Conference Series in Applied Mathematics,No.26.
[20]David Gottlieb,M.Yousuff Hussaini,and Steven A.Orszag.Theory and applications of spectral methods.In Spectral methods for partial differential equations(Hampton,Va.,1982),pages 1-54.SIAM,Philadelphia,PA,1984.
[21]Claudio Canuto,M.Yousuff Hussaini,Alfio Quarteroni,and Thomas A.Zang.Spectral methods in fluid dynamics.Springer Series in Computational Physics.Springer-Verlag,New York,1988.
[22]Bertrand Mercier.An introduction to the numerical analysis of spectral methods,volume 318of Lecture Notes in Physics.Springer-Verlag,Berlin,1989.Translated from the French,edited and with a preface by Nessan Mac Giolla Mhuiris and Mohammed Yousuff Hussaini.
[23]Bengt Fornberg.A practical guide to pseudospectral methods,volume 1 of Cambridge Monographs on Applied and Computational Mathematics.Cambridge University Press,Cambridge,1996.
[24]Christine Bernardi and Yvon Maday.Spectral methods.In Handbook of numerical analysis,Vol.V,Handb.Numer.Anal.,V,pages 209-485.North-Holland,Amsterdam,1997.
[25]Ben-Yu Guo.Spectral methods and their applications.World Scientific Publishing Co.Inc.,River Edge,NJ,1998.
[26]Lloyd N.Trefethen.Spectral methods in MATLAB,volume 10 of Software,Environments,and Tools.Society for Industrial and Applied Mathematics(SIAM),Philadelphia,PA,2000.
[27]John P.Boyd.Chebyshev and Fourier spectral methods.Dover Publications Inc.,Mineola,NY,second edition,2001.
[28]R.L.James and J.A.C.Weideman.Pseudospectral methods for the Benjamin-Ono equation.In R.Vichnevetsky,D.Knight,and G.Richter,editors,Advances in Computer Methods for.Partial Differential Equations,volume ⅶ,pages 371-377.IMACs,Brunswick N.J.,1992.
[29]T.Miloh,M.Prestin,L.Shtilman,and M.P.Tulin.A note on the numerical and N-soliton solutions of the Benjamin-Ono evolution equation.Wave Motion,17:1-10,1993.
[30]B.Pelloni and V.A.Dougalis.Numerical solution of some nonlocal,nonlinear dispersive wave equations.J.Nonlinear Sci.,10:1-22,2000.
[31]B.Pelloni and V.A.Dougalis.Error estimates for a fully discrete spectral scheme for a class of nonlinear,nonlocal dispersive wave equations.Appl.Numer.Math.,37:95-107,2001.
[32]V.Thomee and A.S.Vasudeva Murthy.A numerical method for the periodic Benjamin-Ono equation.BIT,38:597-611,1998.
[33]N.J.Zabusky and M.D.Kruskal.Interaction of "solitions" in a collisionless plasma and the recurrence of initial states.Phys.Reu.Lett.,15:240-243,1965.
[34]Ping Fu Zhao and Meng Zhao Qin.Multisymplectic geometry and multisymplectic Preissmann scheme for the KdV equation.J.Phys.A,33(18):3613-3626,2000.
[35]Yu Shun Wang,Bin Wang,and Chen Xin.Multisymplectic euler box scheme for the KdV equation.Chin.Phys.Lett,24(2):312-314,2007.
[36]Yanfen Cui and Mao De-kang.Numerical method satisfying the first two conservation laws for Korteweg-de Vries equation.Journal of Computational Physics,Received,2007.
[37]F.Tappert.Numerical solution of the korteweg-de vries equation and its step fourier method.In A.C.Newell,editor,Nonlinear Wave Motion,Lectures in Applied Math.,volume 15,pages 215-216.Amer.Math.Soc.,Providence,R.I.,1974.
[38]Hans Schamel and Klaus Els(a|¨)sser.The application of the spectral method to nonlinear wave propagation.J.Computational Phys.,22(4):501-516,1976.
[39]Jose Canosa and Jen(o|¨) Gazdag.The Korteweg-de Vries-Burgers equation.J.Computational Phys.,23(4):393-403,1977.
[40]B.Fornberg and G.B.Whitham.A numerical and theoretical study of certain nonlinear phenomena.Phil.Trans.Roy.Soc.London Ser.A,289:373-404,1978.
[41]K.Abe and O.Inoue.Fourier expansion solution of the KdV equation.J.Comput.Phys.,34:202-210,1980.
[42]P.-Y.Kuo.Error estimations of the spectral method for solving k.d.v.-burgers equation.Acta Mathematica Sinica,28:1-15,1985.
[43]T.F.Chan and T.Kerkhoven.Fourier methods with extended stability intervals for the Korteweg-de Vries equation.SIAM J.Numer.Anal.,22:441-454,1985.
[44]H.-P.Ma and B.-Y.Guo.The Fourier pseudospectral method with a restrain operator for the Korteweg-de Vries equation.J.Comput.Phys.,65:120-137,1986.
[45]Y.Maday and A.Quarteroni.Error analysis for spectral approximation of the Korteweg-de Vries equation.RAIRO Model.Math.Anal.Numer.,22:499-529,1988.
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[49]C.Canuto and A.Quarteroni.Approximation results for orthogonal polynomials in Sobolev space.Math.Comp.,38:67-82,1982.
[50]Heinz-Otto Kreiss and Joseph Oliger.Stability of the Fourier method.SIAM J.Numer.Anal.,16(3):421-433,1979.
[51]R.Iorio and V.de M.Iorio.Fourier analysis and partial differential equations.Number 70in Cambridge Studies in Advanced Mathematics.Cambridge University Press,Cambridge,2001.
[52]H.-P.Ma and W.-W.Sun.Optimal error estimates of the Legendre-Petrov-Galerkin method for the Korteweg-de Vries equation.SIAM J.Numer.Anal.,39:1380-1394,2001.
[53]Lars B.Wahlbin.A dissipative Galerkin method for the numerical solution of first order hyperbolic equations.In C.de Boor,editor,Mathemalical Aspects of finite elements in partial differential equations,pages 147-169.Academic Press,New York,1974.
[2]Hiroaki Ono.Algebraic solitary waves in stratified fluids.J.Phys.Soc.Japan,39(4):1082-1091,1975.
[3]Rafael Jose Iorio,Jr.On the Cauchy problem for the Benjamin-Ono equation.Comm.Partial Differential Equations,11(10):1031-1081,1986.
[4]K.M.Case.Properties of the Benjamin-Ono equation.J.Math.Phys.,20(5):972-977,1979.
[5]L.Abdelouhab,J.L.Bona,M.Felland,and J.-C.Saut.Nonlocal models for nonlinear,dispersive waves.Phys.D,40:360-392,1989.
[6]M.J.Ablowitz and P.A.Clarkson.Salitons,nonlinear evolution equations and inverse scattering,volume 149 of London Mathematical Society Lecture Note Series.Cambridge University Press,Cambridge,1991.
[7]Elias M.Stein.Harmonic analysis:real-variable methods,orthogonality,and oscillatory integrals,volume 43 of Princeton Mathematical Series.Princeton University Press,Princeton,NJ,1993.With the assistance of Timothy S.Murphy,Monographs in Harmonic Analysis,Ⅲ.
[8]R.I.Joseph.Solitary waves in a finite depth fluid.J.Phys.A,10(12):225-227,1977.
[9]T.Kubota and D.R.S.Ko.Weakly nonlinear,long internal gravity waves in stratified fluids of finite depth.AIAA J.Hydronautics,12:157-165,1978.
[10]M.J.Ablowitz,A.S.Fokas,J.Satsuma,and H.Segur.On the periodic intermediate long wave equation.J.Phys.A:Math.Gen.,15:781-786,1982.
[11]P.M.Santini,M.J.Ablowitz,and A.S.Fokas.On the limit from the intermediate long wave equation to the Benjamin-Ono equation.J.Math.Phys.,25(4):892-899,1984.
[12]C.S.Gardner and G.K.Morikawa.Similarity in the asymptotic behaviour of collision-free hydromagnetic waves and water waves.Technical Report Technical Report NYO - 9082,Courant Institute of Mathematical Sciences,New York University,New York,1960.
[13]A.Jeffrey and T.Kakutani.Weak nonlinear dispersive waves:A discussion centered around the Korteweg-de Vries equation.SIAM Rev.,14:582-643,1972.
[14]Robert M.Miura.The Korteweg-de Vries equation:a survey of results.SIAM Rev.,18(3):412-459,1976.
[15]Mark J.Ablowitz and Harvey Segur.Solitons and the inverse scattering transform,volume 4 of SIAM Studies in Applied Mathematics.Society for Industrial and Applied Mathematics (SIAM),Philadelphia,Pa.,1981.
[16]Alan C.Newell.Solitons in mathematics and physics,volume 48 of CBMS-NSF Regional Conference Series in Applied Mathematics.Society for Industrial and Applied Mathematics (SIAM),Philadelphia,PA,1985.
[17]James W.Cooley and John W.Tukey.An algorithm for the machine calculation of complex Fourier series.Math.Comp.,19:297-301,1965.
[18]Steven A.Orszag.Numerical simulation of incompressible flows within simple boundaries.I.Galerkin(spectral) representations.Studies in Appl.Math.,50:293-327,1971.
[19]David Gottlieb and Steven A.Orszag.Numerical analysis of spectral methods:theory and applications.Society for Industrial and Applied Mathematics,Philadelphia,Pa.,1977.CBMS-NSF Regional Conference Series in Applied Mathematics,No.26.
[20]David Gottlieb,M.Yousuff Hussaini,and Steven A.Orszag.Theory and applications of spectral methods.In Spectral methods for partial differential equations(Hampton,Va.,1982),pages 1-54.SIAM,Philadelphia,PA,1984.
[21]Claudio Canuto,M.Yousuff Hussaini,Alfio Quarteroni,and Thomas A.Zang.Spectral methods in fluid dynamics.Springer Series in Computational Physics.Springer-Verlag,New York,1988.
[22]Bertrand Mercier.An introduction to the numerical analysis of spectral methods,volume 318of Lecture Notes in Physics.Springer-Verlag,Berlin,1989.Translated from the French,edited and with a preface by Nessan Mac Giolla Mhuiris and Mohammed Yousuff Hussaini.
[23]Bengt Fornberg.A practical guide to pseudospectral methods,volume 1 of Cambridge Monographs on Applied and Computational Mathematics.Cambridge University Press,Cambridge,1996.
[24]Christine Bernardi and Yvon Maday.Spectral methods.In Handbook of numerical analysis,Vol.V,Handb.Numer.Anal.,V,pages 209-485.North-Holland,Amsterdam,1997.
[25]Ben-Yu Guo.Spectral methods and their applications.World Scientific Publishing Co.Inc.,River Edge,NJ,1998.
[26]Lloyd N.Trefethen.Spectral methods in MATLAB,volume 10 of Software,Environments,and Tools.Society for Industrial and Applied Mathematics(SIAM),Philadelphia,PA,2000.
[27]John P.Boyd.Chebyshev and Fourier spectral methods.Dover Publications Inc.,Mineola,NY,second edition,2001.
[28]R.L.James and J.A.C.Weideman.Pseudospectral methods for the Benjamin-Ono equation.In R.Vichnevetsky,D.Knight,and G.Richter,editors,Advances in Computer Methods for.Partial Differential Equations,volume ⅶ,pages 371-377.IMACs,Brunswick N.J.,1992.
[29]T.Miloh,M.Prestin,L.Shtilman,and M.P.Tulin.A note on the numerical and N-soliton solutions of the Benjamin-Ono evolution equation.Wave Motion,17:1-10,1993.
[30]B.Pelloni and V.A.Dougalis.Numerical solution of some nonlocal,nonlinear dispersive wave equations.J.Nonlinear Sci.,10:1-22,2000.
[31]B.Pelloni and V.A.Dougalis.Error estimates for a fully discrete spectral scheme for a class of nonlinear,nonlocal dispersive wave equations.Appl.Numer.Math.,37:95-107,2001.
[32]V.Thomee and A.S.Vasudeva Murthy.A numerical method for the periodic Benjamin-Ono equation.BIT,38:597-611,1998.
[33]N.J.Zabusky and M.D.Kruskal.Interaction of "solitions" in a collisionless plasma and the recurrence of initial states.Phys.Reu.Lett.,15:240-243,1965.
[34]Ping Fu Zhao and Meng Zhao Qin.Multisymplectic geometry and multisymplectic Preissmann scheme for the KdV equation.J.Phys.A,33(18):3613-3626,2000.
[35]Yu Shun Wang,Bin Wang,and Chen Xin.Multisymplectic euler box scheme for the KdV equation.Chin.Phys.Lett,24(2):312-314,2007.
[36]Yanfen Cui and Mao De-kang.Numerical method satisfying the first two conservation laws for Korteweg-de Vries equation.Journal of Computational Physics,Received,2007.
[37]F.Tappert.Numerical solution of the korteweg-de vries equation and its step fourier method.In A.C.Newell,editor,Nonlinear Wave Motion,Lectures in Applied Math.,volume 15,pages 215-216.Amer.Math.Soc.,Providence,R.I.,1974.
[38]Hans Schamel and Klaus Els(a|¨)sser.The application of the spectral method to nonlinear wave propagation.J.Computational Phys.,22(4):501-516,1976.
[39]Jose Canosa and Jen(o|¨) Gazdag.The Korteweg-de Vries-Burgers equation.J.Computational Phys.,23(4):393-403,1977.
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