Numerical Methods For Several Multi-dimensional Nonlinear Schr(?)dinger-Type Equations

In this thesis,the pseudo-spectral method and the finite difference method are appiled for investigating numerical solutions of several nonlinear Schr(?)dinger/Gross-Pitaevskii equations.Several stable and efficient numerical methods are proposed,and the optimal error estimation are established.Numerical examples are carried out to verify the feasibility of the proposed numerical methods.The main contents of this thesis are summarized as follows:Firstly,the initial-boundary value problem of the general nonlinear Schr(?)dinger equation is numericaly studied.A new Sine pseudo-spectral method is constructed.Then,by using various analytical techniques such as standard energy method,mathematical induction method,matrix analysis and several inverse inequalities,the optimal error estimate is established without any restriction on grid ratio and the size of the initial value.Secondly,the initial-boundary value problem of the nonlinear time-fractional Schr(?)dinger equation is considered.By introducing the L2-1_σformula to discretize Caputo time-fractional derivative,and using the symmetric finite difference scheme to discretize the other terms,a second-order finite difference scheme is proposed.Then,by using the standard energy method,the mathematical induction method and the‘lifting’technique,the optimal H~2error bound is built without any restriction on grid ratio.Thirdly,the initial-boundary value problem of the coupled Gross-Pitaevskii equation is numericaly studied.An implicit finite difference scheme is designed,and it is proved to be uniquely solvable and preserve the mass and energy in the discrete sense.Then,by using the standard energy method,the smooth truncation technique and the‘lifting’technique,the optimal L~∞error estimate is established without any restriction on grid ratio.In addition,a fractional discrete Gronwall inequaty playing a key role in numerical analysis is proved in detail.