带三次项的四阶NLS方程的多辛算法
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摘要
非线性Schr?dinger(NLS)方程在量子力学、等离子体物理、地震学、声学等许多学科中有着广泛的应用。本文研究了带三次项的四阶NLS方程的多辛算法,构造了该方程的多辛Preissman格式、多辛Fourier拟谱格式和分裂多辛格式。
第一章为引言部分,简单介绍了辛算法的发展历史和现状,以及学者在这一领域做的一些研究工作及成果。并且介绍了分裂方法的思想,提出了分裂多辛算法。
第二章首先研究了带三次项的四阶NLS,分析了它的守恒律,并回顾了多辛Hamilton系统的预备知识。
第三章主要讨论了带三次项的四阶NLS的多辛算法,构造了它的多辛Preissman格式,证明了此多辛格式保持电荷守恒。最后用数值实验验证了该格式具有长时间的数值稳定性。
第四章首先简单介绍了多辛Fourier拟谱方法的预备知识,构造了带三次项的四阶NLS的多辛Fourier拟谱格式,证明了它的电荷守恒律,并通过数值实验验证了该格式具有长时间的模拟能力。
第五章首先介绍了时间分裂方法的思想,然后构造了带三次项的四阶NLS的分裂多辛格式,分析了该格式的稳定性。在数值实验中,对格式的有效性进行了验证,即格式不仅具有长时间的数值计算能力而且效率很高。
Many physical phenomena in quantum mechanics, plasma astrophysics, seismology, acoustics and so on can be described by the Schr?dinger-type equations.In this thesis, we study multi-symplectic algorithms for the fourth-order Schr?dinger equation with cubic nonlinear term, and we construct multi-symplectic Preissman scheme, multi-symplectic Fourier pseudo-spectral scheme, and time-splitting multi-symplectic scheme.
In Chapter 1, we simply introduce the history and present status of symplectic algorithm, and numbers of related achievements in this field. Moreover, we introduce the time-splitting method and present the time-splitting multi-symplectic algorithm in this chapter.
In Chapter 2, first, we investigate the fourth-order Schr?dinger equation with cubic nonlinear term, analyze its conservation laws and present the preliminary knowledge for the multi-symplectic Hamiltonian system.
In Chapter 3, we study multi-symplectic algorithm, mainly on its multi-symplectic Preissman scheme for the fourth-order Schr?dinger equation with cubic nonlinear term. It is proved that the multi-symplectic scheme preserves the charge conservation law. Furthermore, it is suggested that it is stable through energy method. Numerical results verify that the scheme is capable of simulating the original over a long time.
In Chapter 4, the multi-symplectic Fourier method is reviewed firstly. Then, it is applied to the fourth-order NLS equations. It is proved that the scheme preserve the charge conservation exactly. Lastly, it is illustrated that the scheme is capable of simulating the original in a long time via numerical experiment.
In Chapter 5, we briefly introduce the time-splitting method, and a time-splitting multi-symplectic scheme is constructed based on the Strang time-splitting method. The unconditional stability is verified by numerical analysis. At last, the results of the numerical examples demonstrate the stability and the efficiency of the scheme.
第一章为引言部分,简单介绍了辛算法的发展历史和现状,以及学者在这一领域做的一些研究工作及成果。并且介绍了分裂方法的思想,提出了分裂多辛算法。
第二章首先研究了带三次项的四阶NLS,分析了它的守恒律,并回顾了多辛Hamilton系统的预备知识。
第三章主要讨论了带三次项的四阶NLS的多辛算法,构造了它的多辛Preissman格式,证明了此多辛格式保持电荷守恒。最后用数值实验验证了该格式具有长时间的数值稳定性。
第四章首先简单介绍了多辛Fourier拟谱方法的预备知识,构造了带三次项的四阶NLS的多辛Fourier拟谱格式,证明了它的电荷守恒律,并通过数值实验验证了该格式具有长时间的模拟能力。
第五章首先介绍了时间分裂方法的思想,然后构造了带三次项的四阶NLS的分裂多辛格式,分析了该格式的稳定性。在数值实验中,对格式的有效性进行了验证,即格式不仅具有长时间的数值计算能力而且效率很高。
Many physical phenomena in quantum mechanics, plasma astrophysics, seismology, acoustics and so on can be described by the Schr?dinger-type equations.In this thesis, we study multi-symplectic algorithms for the fourth-order Schr?dinger equation with cubic nonlinear term, and we construct multi-symplectic Preissman scheme, multi-symplectic Fourier pseudo-spectral scheme, and time-splitting multi-symplectic scheme.
In Chapter 1, we simply introduce the history and present status of symplectic algorithm, and numbers of related achievements in this field. Moreover, we introduce the time-splitting method and present the time-splitting multi-symplectic algorithm in this chapter.
In Chapter 2, first, we investigate the fourth-order Schr?dinger equation with cubic nonlinear term, analyze its conservation laws and present the preliminary knowledge for the multi-symplectic Hamiltonian system.
In Chapter 3, we study multi-symplectic algorithm, mainly on its multi-symplectic Preissman scheme for the fourth-order Schr?dinger equation with cubic nonlinear term. It is proved that the multi-symplectic scheme preserves the charge conservation law. Furthermore, it is suggested that it is stable through energy method. Numerical results verify that the scheme is capable of simulating the original over a long time.
In Chapter 4, the multi-symplectic Fourier method is reviewed firstly. Then, it is applied to the fourth-order NLS equations. It is proved that the scheme preserve the charge conservation exactly. Lastly, it is illustrated that the scheme is capable of simulating the original in a long time via numerical experiment.
In Chapter 5, we briefly introduce the time-splitting method, and a time-splitting multi-symplectic scheme is constructed based on the Strang time-splitting method. The unconditional stability is verified by numerical analysis. At last, the results of the numerical examples demonstrate the stability and the efficiency of the scheme.
引文
[1] K. Feng. On difference schemes and symplectic geometry, Proceeding of the 1984 Beijing symposium on differential geometry and differential equations[C]. Beijing: Science Press, 1985, 42-58.
[2] K. Feng. Difference schemes of Hamilton formalism and symplectic geometry[J]. J Comput. Math., 1986, 4(3): 279-289.
[3] K. Feng.Collected works of Feng Kang (Ⅱ)[C]. Beijing: National Defense Industry Press, 1995.
[4]冯康,秦孟兆.哈密顿系统的辛几何算法[M].杭州:浙江科学技术出版社,2002.
[5]秦孟兆.辛几何及计算哈密顿力学[J].力学与实践,1990,12(6): 1-20.
[6]秦孟兆.任意阶精度的蛙跳格式的稳定性分析[J].计算数学,1992, 14(1): 1-9.
[7] J. L. Hong , Y. Liu. Multisymplecticity of the central box scheme for a class of Hamiltonian PDEs and an application to quasi-periodically solitary waves[J]. Math. Comput. Model., 2004,39: 1035-1047.
[8] J. L. Hong , Y. Liu. A novel numerical approach to simulating nonlinear Schr?dinger equation with variable coefficients [J]. Appl. Math. Letters, 2002, 16: 759-765.
[9] T. J. Bridges, S. Reich. Multi-symplectic spectral discretization for the Zakharov- Kaznetsov and shallow water equation [J]. Physica D, 2001:491-504.
[10] S. Reich. Finite volume methods for multi-symplectic PDEs[J]. BIT, 2000, 40(3): 559-582.
[11] J. L. Hong, M. Z. Qin. Multi-sympliecticity of centered box discretizations for Hamiltonian PDEs with m≥2 space dimensions [J]. Appl. Math. Letters, 2002, 15: 1005 -1011.
[12] M. Z. Qin. Multi-stage symplectic schemes of two kind of Hamiltonian systems of wave equations[J]. J. Comput. Math. Appl., 1990, 19: 51-62.
[13]孔令华.一些非线性发展方程(组)的辛和多辛算法[D].中国科学技术大学博士学位论文,2007年5月.
[14]孙志忠.偏微分方程的数值解法[M].北京:科学出版社,2005年.
[15]余德浩,汤华中.微分方程数值解法[M].科学出版社,北京,2004年.
[16] G. B. Yu, Pascual P J., Rodriguez M J, Luis Vazquez. Numerical Solution of the Sine- Gordon Equation[J]. Appl. Math. Comput., 1986, 18: 1-14.
[17]常谦顺.一类非线性NLS的守恒差分格式[J].科学通报,1981, 26: 1094-1097.
[18] L. H. Kong, J. L. Hong. Symplectic integrator for nonlinear high order Schr?dinger equation with a trapped term[J]. Comput. Math. Appl., 2009, 231: 664-679.
[19]王廷春,郭柏灵.一维非线性Schr?dinger方程的两个无条件收敛的守恒紧致差分格式[J].中国科学(数学), 2011, 41(3): 207-233.
[20]曾文平.高阶Schr?dinger方程的高精度辛格式[J].计算物理,2004,21(2): 106-110.
[21]曾文平.高阶Schr?dinger方程的哈密顿型蛙跳格式[J].高等学校计算数学学报,1995,17(4): 305-317.
[22] A. Zisowsky, M. Ehrhardt. Discrete artificial boundary conditions for nonlinear Schr?dinger equations[J]. Math. Comput. Model., 2008, 47: 1264-1283.
[23] J. A. C. Weideman. Spectral differentiation matrices for the numerical solution of Schr?dinger’s equation[J]. J. Phys. A: Math. Gen., 2006, 39: 10229-10237.
[24]孔令华,曾文平. SRLW方程的多辛Fourier谱格式及其守恒律[J].计算物理,2006,23(1): 25-31.
[25]王兰.多辛Preissman格式及其应用[J].江西师范大学学报,2009,33(1):42-46.
[26]王雨顺,王斌,秦孟兆. 2+1维Sine-Gordon方程的多辛格式的复合构造[J].中国科学: A. 2003, 33(3): 272-281.
[27]向新民.谱方法的数值分析[M].北京:科学出版社,2000年.
[28] Orszag. Steven A. Comparison of pseudo-spectral and spectral approximations study and application of math[J]. Stud. Appl. Math., 1972, 51: 253-259.
[29] Anton Arnold, Christian Ringhofer. An operator splitting method for the Wigner Poisson problem[J]. SIAM J. Numer. Anal., 1996, 33(4):1622-1643.
[30] T. Tang. Convergence analysis for operator-splitting methods applied to conservation laws with stiff source terms. SIAM J. Numer. Anal., 1998, 35(5): 1939-1968.
[31] Randall J. LeVeque. Intermediate boundary conditions for time-split methods applied to hyperbolic partial differential equations[J]. Math. Comput., 1986, 47(175): 37-54.
[32] L. H. Kong, J. L. Hong, J. Zhang. Splitting multi-symplectic method for Maxwell's equation[J]. J. Comput. Phys. 2010, 229: 4259-4278.
[33] W. Z. Bao, J. Shen. A fourth-order time-splitting Laguerre-Hermite pseudo-spectral method for Bose-Einstein condensates [J]. SIAM J. Sci. Comput., 2005, 26(6): 2010- 2028.
[34] Q. S. Chang, Y. S. Wong, C. K. Lin. Numerical computations for long-wave short- wave interaction equations in semi-classical limit[J]. J. Comput. Phys., 2008, 227: 8489-8507.
[35] H. Q. Wang. A time-splitting spectral method for coupled Gross-Pitaevskii equation with applications to rotating Bose-Einstein condensates[J]. J. Comput. Appl. Math., 2007, 205:88-104.
[36]王廷春.某些非线性孤立波方程(组)的数值算法研究[D].南京航空航天大学博士学位论文. 2008年3月.
[37] J.A.C. Weideman, B.M. Herbst. Split-step methods for the solution of the non-linear Schr?dinger equations[J]. SIAM J. Numer. Anal., 1986, 23(3): 485-507.
[38] R. Zhang, K. Zhang, Y.S. Zhou. Numerical study of time-splitting and space-time adaptive wavelet scheme for Schr?dinger equations[J]. J. Comput. Appl. Math., 2006, 195: 263-273.
[39] Randall J. LeVeque. Intermediate boundary conditions for time-spliting methods applied to hyperbolic partial differential equations[J]. Math. Comput., 1986, 47(175): 37-54.
[40] Abdul-Majid Wazwaz. Exact solutions for the fourth order nonlinear Schr?dinger equations with cubic and power law nonlinearities[J]. Math. Comput. Model., 2006, 43: 802-808.
[41] H. Q. Wang. Numerical studies on the split-step finite difference method for nonlinear Schr?dinger equations[J]. Appl. Math. Comput., 2005, 170: 17-35.
[42] C. A. J. Fletcher. Time-splitting and the finite element method[J]. University of Sydney, Sydney, NSW, 2006, 37-47.
[43] G. M. Muslu, H. A. Erbay. Higher-order split-step Fourier schemes for the generalized nonlinear Schr?dinger equation[J]. Math. Comput., 2005, 67: 581-595.
[44] T Sorevik, L B Madsen, J P Hansen. A spectral method for integration of the time-dependent Schr?dinger equation in hyper spherical coordinates[J]. J. Phys. A: Math. Gen., 2005, 38: 6977-6985.
[2] K. Feng. Difference schemes of Hamilton formalism and symplectic geometry[J]. J Comput. Math., 1986, 4(3): 279-289.
[3] K. Feng.Collected works of Feng Kang (Ⅱ)[C]. Beijing: National Defense Industry Press, 1995.
[4]冯康,秦孟兆.哈密顿系统的辛几何算法[M].杭州:浙江科学技术出版社,2002.
[5]秦孟兆.辛几何及计算哈密顿力学[J].力学与实践,1990,12(6): 1-20.
[6]秦孟兆.任意阶精度的蛙跳格式的稳定性分析[J].计算数学,1992, 14(1): 1-9.
[7] J. L. Hong , Y. Liu. Multisymplecticity of the central box scheme for a class of Hamiltonian PDEs and an application to quasi-periodically solitary waves[J]. Math. Comput. Model., 2004,39: 1035-1047.
[8] J. L. Hong , Y. Liu. A novel numerical approach to simulating nonlinear Schr?dinger equation with variable coefficients [J]. Appl. Math. Letters, 2002, 16: 759-765.
[9] T. J. Bridges, S. Reich. Multi-symplectic spectral discretization for the Zakharov- Kaznetsov and shallow water equation [J]. Physica D, 2001:491-504.
[10] S. Reich. Finite volume methods for multi-symplectic PDEs[J]. BIT, 2000, 40(3): 559-582.
[11] J. L. Hong, M. Z. Qin. Multi-sympliecticity of centered box discretizations for Hamiltonian PDEs with m≥2 space dimensions [J]. Appl. Math. Letters, 2002, 15: 1005 -1011.
[12] M. Z. Qin. Multi-stage symplectic schemes of two kind of Hamiltonian systems of wave equations[J]. J. Comput. Math. Appl., 1990, 19: 51-62.
[13]孔令华.一些非线性发展方程(组)的辛和多辛算法[D].中国科学技术大学博士学位论文,2007年5月.
[14]孙志忠.偏微分方程的数值解法[M].北京:科学出版社,2005年.
[15]余德浩,汤华中.微分方程数值解法[M].科学出版社,北京,2004年.
[16] G. B. Yu, Pascual P J., Rodriguez M J, Luis Vazquez. Numerical Solution of the Sine- Gordon Equation[J]. Appl. Math. Comput., 1986, 18: 1-14.
[17]常谦顺.一类非线性NLS的守恒差分格式[J].科学通报,1981, 26: 1094-1097.
[18] L. H. Kong, J. L. Hong. Symplectic integrator for nonlinear high order Schr?dinger equation with a trapped term[J]. Comput. Math. Appl., 2009, 231: 664-679.
[19]王廷春,郭柏灵.一维非线性Schr?dinger方程的两个无条件收敛的守恒紧致差分格式[J].中国科学(数学), 2011, 41(3): 207-233.
[20]曾文平.高阶Schr?dinger方程的高精度辛格式[J].计算物理,2004,21(2): 106-110.
[21]曾文平.高阶Schr?dinger方程的哈密顿型蛙跳格式[J].高等学校计算数学学报,1995,17(4): 305-317.
[22] A. Zisowsky, M. Ehrhardt. Discrete artificial boundary conditions for nonlinear Schr?dinger equations[J]. Math. Comput. Model., 2008, 47: 1264-1283.
[23] J. A. C. Weideman. Spectral differentiation matrices for the numerical solution of Schr?dinger’s equation[J]. J. Phys. A: Math. Gen., 2006, 39: 10229-10237.
[24]孔令华,曾文平. SRLW方程的多辛Fourier谱格式及其守恒律[J].计算物理,2006,23(1): 25-31.
[25]王兰.多辛Preissman格式及其应用[J].江西师范大学学报,2009,33(1):42-46.
[26]王雨顺,王斌,秦孟兆. 2+1维Sine-Gordon方程的多辛格式的复合构造[J].中国科学: A. 2003, 33(3): 272-281.
[27]向新民.谱方法的数值分析[M].北京:科学出版社,2000年.
[28] Orszag. Steven A. Comparison of pseudo-spectral and spectral approximations study and application of math[J]. Stud. Appl. Math., 1972, 51: 253-259.
[29] Anton Arnold, Christian Ringhofer. An operator splitting method for the Wigner Poisson problem[J]. SIAM J. Numer. Anal., 1996, 33(4):1622-1643.
[30] T. Tang. Convergence analysis for operator-splitting methods applied to conservation laws with stiff source terms. SIAM J. Numer. Anal., 1998, 35(5): 1939-1968.
[31] Randall J. LeVeque. Intermediate boundary conditions for time-split methods applied to hyperbolic partial differential equations[J]. Math. Comput., 1986, 47(175): 37-54.
[32] L. H. Kong, J. L. Hong, J. Zhang. Splitting multi-symplectic method for Maxwell's equation[J]. J. Comput. Phys. 2010, 229: 4259-4278.
[33] W. Z. Bao, J. Shen. A fourth-order time-splitting Laguerre-Hermite pseudo-spectral method for Bose-Einstein condensates [J]. SIAM J. Sci. Comput., 2005, 26(6): 2010- 2028.
[34] Q. S. Chang, Y. S. Wong, C. K. Lin. Numerical computations for long-wave short- wave interaction equations in semi-classical limit[J]. J. Comput. Phys., 2008, 227: 8489-8507.
[35] H. Q. Wang. A time-splitting spectral method for coupled Gross-Pitaevskii equation with applications to rotating Bose-Einstein condensates[J]. J. Comput. Appl. Math., 2007, 205:88-104.
[36]王廷春.某些非线性孤立波方程(组)的数值算法研究[D].南京航空航天大学博士学位论文. 2008年3月.
[37] J.A.C. Weideman, B.M. Herbst. Split-step methods for the solution of the non-linear Schr?dinger equations[J]. SIAM J. Numer. Anal., 1986, 23(3): 485-507.
[38] R. Zhang, K. Zhang, Y.S. Zhou. Numerical study of time-splitting and space-time adaptive wavelet scheme for Schr?dinger equations[J]. J. Comput. Appl. Math., 2006, 195: 263-273.
[39] Randall J. LeVeque. Intermediate boundary conditions for time-spliting methods applied to hyperbolic partial differential equations[J]. Math. Comput., 1986, 47(175): 37-54.
[40] Abdul-Majid Wazwaz. Exact solutions for the fourth order nonlinear Schr?dinger equations with cubic and power law nonlinearities[J]. Math. Comput. Model., 2006, 43: 802-808.
[41] H. Q. Wang. Numerical studies on the split-step finite difference method for nonlinear Schr?dinger equations[J]. Appl. Math. Comput., 2005, 170: 17-35.
[42] C. A. J. Fletcher. Time-splitting and the finite element method[J]. University of Sydney, Sydney, NSW, 2006, 37-47.
[43] G. M. Muslu, H. A. Erbay. Higher-order split-step Fourier schemes for the generalized nonlinear Schr?dinger equation[J]. Math. Comput., 2005, 67: 581-595.
[44] T Sorevik, L B Madsen, J P Hansen. A spectral method for integration of the time-dependent Schr?dinger equation in hyper spherical coordinates[J]. J. Phys. A: Math. Gen., 2005, 38: 6977-6985.
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