带五次项的非线性Schr（？）dinger方程的辛和多辛算法
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摘要
1984年我国计算数学的奠基人冯康院士首次系统地提出了能保持哈密顿系统辛结构不变的辛几何算法.近几年来,此算法得到了迅猛的发展,并成功地解决了许多实际问题,模拟了各种物理现象.
本文考虑了带五次项的非线性Schr?dinger方程的(单)辛和多辛算法.构造了该方程的(单)辛格式、多辛Preissman格式和多辛Fourier拟谱格式.
本文第一章为引言部分,简单介绍了辛算法的发展历史和现状,以及在这一领域取得的一些研究成果.
第二章主要讨论了带五次项的非线性Schr?dinger方程的(单)辛算法,构造了它的(单)辛格式,并用数值实验验证了该格式具有长时间的数值模拟能力.
第三章首先简单介绍了多辛哈密顿系统及其相关的守恒律,然后构造了带五次项非线性Schr?dinger方程的多辛Preissman格式,证明了此多辛格式保持电荷守恒,分析了它的能量误差,并用能量方法分析了该格式的稳定性和收敛性.最后用数值实验验证了我们的理论分析是正确的,即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,并构造了带五次项的非线性Schr?dinger方程的多辛Fourier拟谱格式,最后通过数值例子说明了该格式的有效性.
第五章对全文的主要内容做了简单的总结,并对(单)辛和多辛算法的未来方向做了一些展望.
In 1984, academician Kang Feng, the founder of Chinese computational mathematics, put forward the symeplectic geometric algorithm systemically which preserves symplectic structure of the Hamiltonian system. In the recent years, the algorithms have solved many practical problems successfully and simulated lots of physical phenomena.
In this paper, we consider (single) symplectic and multi-symplectic algorithms for nonlinear Schr?dinger equation involving quintic term, and we construct (single) symplectic scheme, multi-symplectic preissman scheme and multi-symplectic Fourier pseudo-spectral scheme for the equation.
In chapter one, we present the abstract, simply introduce the history and present state of symplectic algorithm, and numbers of related achievements in this field.
In chapter two, we consider symplectic algorithm and construct (single) symplectic scheme for nonlinear Schr?dinger equation involving quintic term, and prove that the symplectic scheme is capable of simulating the original in a long time by numerical experiment.
In chapter three, firstly, we simply present the multi-symplectic Hamiltonian system and its associated conservation laws. Secondly, we construct the multi-symplectic preissman scheme for the NLS equation, prove the scheme preserve the charge conservation law exactly and analyze the energy residual, and we prove the stability and convergence of the multi-symplectic scheme by energy method. Lastly, the numerrical experiment shows that the theoretical analysis is correct,the multi-symplectic scheme is capable of simulating the original in a long time.
In chapter four, we present the prepare knowledge of the multi-symplectic Fourier method, then construct the multi-symplectic Fourier pesudo-spetral scheme for the NLS equation, lastly we prove that the scheme is reliable by numerical experiment.
In the last chapter, we summarize the main conclusions of the dissertation, and look into the future prospects of the (single) symplectic and multi-symplectic algorithms.
本文考虑了带五次项的非线性Schr?dinger方程的(单)辛和多辛算法.构造了该方程的(单)辛格式、多辛Preissman格式和多辛Fourier拟谱格式.
本文第一章为引言部分,简单介绍了辛算法的发展历史和现状,以及在这一领域取得的一些研究成果.
第二章主要讨论了带五次项的非线性Schr?dinger方程的(单)辛算法,构造了它的(单)辛格式,并用数值实验验证了该格式具有长时间的数值模拟能力.
第三章首先简单介绍了多辛哈密顿系统及其相关的守恒律,然后构造了带五次项非线性Schr?dinger方程的多辛Preissman格式,证明了此多辛格式保持电荷守恒,分析了它的能量误差,并用能量方法分析了该格式的稳定性和收敛性.最后用数值实验验证了我们的理论分析是正确的,即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,并构造了带五次项的非线性Schr?dinger方程的多辛Fourier拟谱格式,最后通过数值例子说明了该格式的有效性.
第五章对全文的主要内容做了简单的总结,并对(单)辛和多辛算法的未来方向做了一些展望.
In 1984, academician Kang Feng, the founder of Chinese computational mathematics, put forward the symeplectic geometric algorithm systemically which preserves symplectic structure of the Hamiltonian system. In the recent years, the algorithms have solved many practical problems successfully and simulated lots of physical phenomena.
In this paper, we consider (single) symplectic and multi-symplectic algorithms for nonlinear Schr?dinger equation involving quintic term, and we construct (single) symplectic scheme, multi-symplectic preissman scheme and multi-symplectic Fourier pseudo-spectral scheme for the equation.
In chapter one, we present the abstract, simply introduce the history and present state of symplectic algorithm, and numbers of related achievements in this field.
In chapter two, we consider symplectic algorithm and construct (single) symplectic scheme for nonlinear Schr?dinger equation involving quintic term, and prove that the symplectic scheme is capable of simulating the original in a long time by numerical experiment.
In chapter three, firstly, we simply present the multi-symplectic Hamiltonian system and its associated conservation laws. Secondly, we construct the multi-symplectic preissman scheme for the NLS equation, prove the scheme preserve the charge conservation law exactly and analyze the energy residual, and we prove the stability and convergence of the multi-symplectic scheme by energy method. Lastly, the numerrical experiment shows that the theoretical analysis is correct,the multi-symplectic scheme is capable of simulating the original in a long time.
In chapter four, we present the prepare knowledge of the multi-symplectic Fourier method, then construct the multi-symplectic Fourier pesudo-spetral scheme for the NLS equation, lastly we prove that the scheme is reliable by numerical experiment.
In the last chapter, we summarize the main conclusions of the dissertation, and look into the future prospects of the (single) symplectic and multi-symplectic algorithms.
引文
[1] Ernst Hairer, Christian Lubich, Gerhard Wanner. Geometric numerical integra- tion [M], Berlin: Springer press, 2001.
[2] K.Feng.Collected works of Feng Kang (Ⅱ)[J], Beijing: National Defense Industry Press, 1995.
[3]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.
[4]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.
[5]丁培柱,李延欣等.量子系统的辛算法[J],吉林大学自然科学学报,1993,4:75-82.
[6]陈景波,秦孟兆.辛几何算法在射线追踪中的应用[J],数值计算与计算机应用,2000,21(4):255-265.
[7]刘林,赵长印等.辛算法在动力天文学中的应用(Ⅲ)[J],天文学报,1994,35(1).
[8] J.L.Hong and Y.Liu. A novel numerical approach to simulating nonlinear Schr?dinger equation with variable coefficients [J]. Appl. Math. Letters, 2002,16:759-765.
[9] J.L.Hong and Y.Liu. Multi-sympliecticity of centered box discretizations for a class of Hamiltonian PDEs and an application to quasi-periodically solitary wave of qpkdv equation[J]. Preprint (2001).
[10] 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.
[11] J.L.Hong, Y.Liu, Hans Munthe-Kass, Antonella Zanna. Globality conservative properties and error estimation of a Multi-symplectic scheme for Schr?dinger equation with variable coefficients [J]. Appl. Math. Letters, preprint (2003).
[12] T.J.Bridges, S. Reich. Multi-symplectic spectral discretization for the Zakharov-Kaznetsov and shallow water equation [J]. Physica D,2001:491-504.
[13] S.Reich. Finite volume methods for multi-symplectic PDEs[J].BIT,2000,40 (3):559-582.
[14]赵平福.KDV方程的多辛算法[D],博士论文(1999),中国科学院数学与系统科院.
[15]蒋长锦.二维Sine-Gordon方程的多辛格式[J],中国科学技术大学学报,2003,33(3):253-261.
[16]蒋长锦.二维非定常Sine-Gordon方程的辛算法及其孤子数值模拟[J],计算物理,2003, 20(4):321-325.
[17]王雨顺,王斌,秦孟兆.2+1维Sine-Gordon方程多辛格式的复合构造[J],中国科学(A), 2003, 33(3):272-281.
[18]王雨顺.孤立波方程的辛和多辛算法的构造和计算(D),博士论文(2001),中国科学院数学与系统科学院.
[19] P.F.Zhao,M.Z.Qi.Approximation for the Kdv equations as a Hamiltonian system[J],Computers Math.Applic.,2000,39:1-11.
[20] P.F.Zhao,M.Z.Qin.Multi-symplectic geometry and Multi-symplectic Preissman schemes for the Kdv equations[J], J.Phys. A Math.Gen.,2000,33:3613-3626.
[21]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.
[22] W.P.Zeng, M.Z.Qin, L.Y.Huang.Construction of Multi-Symplectic scheme for“good”Boussinesq equation [J], CCAST-WL WORKSHOP, 1999,6(1):231-245.
[23] L.Y.Huang, W.P.Zeng, M.Z.Qin.A new Multi-Symplectic scheme for nonlinear“good”Boussinesq equation [J], CCAST-WL WORKSHOP, 1999, 6(1):107-123.
[24]单双荣.梁振动方程的多辛Fourier拟谱算法[J],华侨大学学报(自然科学版),2006,27(3):234-237.
[25]孔令华,曾文平,刘儒勋,孔令健.SRLW方程的多辛Fourier谱格式及其守恒律[J],2006, [J],2006,23:25-31.
[26]张鲁明,常谦顺.非线性Schr?dinger方程的一个守恒差分格式[J],高等应用数学学报,2000,23(3):351-358.
[27]鲁百年.一类非自共轭非线性Schr?dinger方程组的有限差分方法[J],计算数学,1989,11(2):118-127.
[28]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.
[29] J.B.Chen, M.Z.Qin.Multi-Symplectic Fourier pseudo-spectral method for the nonlinear Schr?dinger equation [J], ETNA, 2001, 12:193-204.
[30] Q.S.Chang, L.B.Xu.A Numerical method for a system of Generalized Nonlinear Schr?dinger equations [J], J.Comput.Math. 1986, 4:191-199.
[31] L.S.Peranich . A Finite Difference Scheme for Solving a Nonlinear Schr?dinger Equation with a Linear Damping Term [J], J.Comput.Phys., 1987,68:501-505.
[32]张鲁明,常谦顺.带五次项的非线性Schr?dinger方程的守恒数值格式[J],应用数学,1999,12(1):65-71.
[33]张鲁明,常谦顺.带五次项的非线性Schr?dinger方程差分解法[J],应用数学学报,2000,23(3):351-358.
[34]常谦顺.一类非线性Schr?dinger方程的守恒差分格式[J],科学通报,1981,26:1094-1097.
[35] Y.L.Zhou.Applications of Discrete Functional Analysis to the Finite Difference Method [M], International Academic Publishers, 1990.
[2] K.Feng.Collected works of Feng Kang (Ⅱ)[J], Beijing: National Defense Industry Press, 1995.
[3]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.
[4]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.
[5]丁培柱,李延欣等.量子系统的辛算法[J],吉林大学自然科学学报,1993,4:75-82.
[6]陈景波,秦孟兆.辛几何算法在射线追踪中的应用[J],数值计算与计算机应用,2000,21(4):255-265.
[7]刘林,赵长印等.辛算法在动力天文学中的应用(Ⅲ)[J],天文学报,1994,35(1).
[8] J.L.Hong and Y.Liu. A novel numerical approach to simulating nonlinear Schr?dinger equation with variable coefficients [J]. Appl. Math. Letters, 2002,16:759-765.
[9] J.L.Hong and Y.Liu. Multi-sympliecticity of centered box discretizations for a class of Hamiltonian PDEs and an application to quasi-periodically solitary wave of qpkdv equation[J]. Preprint (2001).
[10] 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.
[11] J.L.Hong, Y.Liu, Hans Munthe-Kass, Antonella Zanna. Globality conservative properties and error estimation of a Multi-symplectic scheme for Schr?dinger equation with variable coefficients [J]. Appl. Math. Letters, preprint (2003).
[12] T.J.Bridges, S. Reich. Multi-symplectic spectral discretization for the Zakharov-Kaznetsov and shallow water equation [J]. Physica D,2001:491-504.
[13] S.Reich. Finite volume methods for multi-symplectic PDEs[J].BIT,2000,40 (3):559-582.
[14]赵平福.KDV方程的多辛算法[D],博士论文(1999),中国科学院数学与系统科院.
[15]蒋长锦.二维Sine-Gordon方程的多辛格式[J],中国科学技术大学学报,2003,33(3):253-261.
[16]蒋长锦.二维非定常Sine-Gordon方程的辛算法及其孤子数值模拟[J],计算物理,2003, 20(4):321-325.
[17]王雨顺,王斌,秦孟兆.2+1维Sine-Gordon方程多辛格式的复合构造[J],中国科学(A), 2003, 33(3):272-281.
[18]王雨顺.孤立波方程的辛和多辛算法的构造和计算(D),博士论文(2001),中国科学院数学与系统科学院.
[19] P.F.Zhao,M.Z.Qi.Approximation for the Kdv equations as a Hamiltonian system[J],Computers Math.Applic.,2000,39:1-11.
[20] P.F.Zhao,M.Z.Qin.Multi-symplectic geometry and Multi-symplectic Preissman schemes for the Kdv equations[J], J.Phys. A Math.Gen.,2000,33:3613-3626.
[21]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.
[22] W.P.Zeng, M.Z.Qin, L.Y.Huang.Construction of Multi-Symplectic scheme for“good”Boussinesq equation [J], CCAST-WL WORKSHOP, 1999,6(1):231-245.
[23] L.Y.Huang, W.P.Zeng, M.Z.Qin.A new Multi-Symplectic scheme for nonlinear“good”Boussinesq equation [J], CCAST-WL WORKSHOP, 1999, 6(1):107-123.
[24]单双荣.梁振动方程的多辛Fourier拟谱算法[J],华侨大学学报(自然科学版),2006,27(3):234-237.
[25]孔令华,曾文平,刘儒勋,孔令健.SRLW方程的多辛Fourier谱格式及其守恒律[J],2006, [J],2006,23:25-31.
[26]张鲁明,常谦顺.非线性Schr?dinger方程的一个守恒差分格式[J],高等应用数学学报,2000,23(3):351-358.
[27]鲁百年.一类非自共轭非线性Schr?dinger方程组的有限差分方法[J],计算数学,1989,11(2):118-127.
[28]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.
[29] J.B.Chen, M.Z.Qin.Multi-Symplectic Fourier pseudo-spectral method for the nonlinear Schr?dinger equation [J], ETNA, 2001, 12:193-204.
[30] Q.S.Chang, L.B.Xu.A Numerical method for a system of Generalized Nonlinear Schr?dinger equations [J], J.Comput.Math. 1986, 4:191-199.
[31] L.S.Peranich . A Finite Difference Scheme for Solving a Nonlinear Schr?dinger Equation with a Linear Damping Term [J], J.Comput.Phys., 1987,68:501-505.
[32]张鲁明,常谦顺.带五次项的非线性Schr?dinger方程的守恒数值格式[J],应用数学,1999,12(1):65-71.
[33]张鲁明,常谦顺.带五次项的非线性Schr?dinger方程差分解法[J],应用数学学报,2000,23(3):351-358.
[34]常谦顺.一类非线性Schr?dinger方程的守恒差分格式[J],科学通报,1981,26:1094-1097.
[35] Y.L.Zhou.Applications of Discrete Functional Analysis to the Finite Difference Method [M], International Academic Publishers, 1990.
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