【作者】 曹莹；

【导师】 孔令华；

【作者基本信息】 江西师范大学 ， 计算数学， 2011， 硕士

【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。更多还原

【Abstract】 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.更多还原