| 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. |
PDF Full Text Request