Square-conservative Scheme For The Nonlinear Schr(?)dinger Equation

The nonlinear Schr(?)dinger equation is the cornerstone of quantum mechanics and has very important applications in many fields.The development of efficient and stable high-order accuracy numerical algorithms for this equation has always been one of the research hotspots in the field of scientific computation.A lot of numerical experiments and theoretical analysis show that numerical algorithms which can preserve some structure of the original differential equations have better performance in numerical stability and long-time simulation than traditional algorithms.Lie group algorithm is a structure-preserving algorithm developed in recent years.This method can effectively construct structure-preserving algorithms for differential equations on manifolds.Its main idea is to make use of the mapping between Lie groups and Lie algebras,to locally rewrite differential equations on manifolds into equations on Lie algebras,and to obtain numerical solutions on original manifolds by solving numerically equations on Lie algebras.At present,Lie group algorithm has been successfully applied to some classical manifold equations and also to a few partial differential equations.Based on the idea of Lie group algorithm,this thesis develops several kinds of square-conservative numerical algorithms for the nonlinear Schr(?)dinger equation.Almost existing structure-preserving algorithms only have second-order accuracy in time.How to obtain higher-order structure-preserving algorithms has always been one of the challenges in the research fields.Here in this thesis,structure-preserving algorithms with high up to fourth-order accuracy can be obtained for the nonlinear Schr(?)dinger equation through the idea of Lie group algorithms.First,we transform the nonlinear Schr(?)dinger equation in the Hamiltonian form and give the corresponding conservation properties.Then,an appropriate finite difference method and the Fourier pseudo-spectral method are selected to spatially discretize the nonlinear Schr(?)dinger equation in space and time respectively.We show that the obtained semi-discrete system is still a conservative system,and the square conserved structure of the original system can be conserved.Afterwards,we use three different Lie group methods,i.e.,C-G method?M-K method and Cayley transformation method,to discrete the obtained system in time respectively,thus a series of full discreted schemes are obtained which are square-conservative schemes for the nonlinear Schr(?)dinger equation.In C-G method and M-K method,we need to compute the matrix exponential which is time consuming especially for the high order algorithms,so we only structure first-order and second-order accuracy algorithms in time.But for the Cayley transformation method,which avoids to compute the matrix exponential,we construct the structure-preserving algorithms with accuracy varied from first order to fourth order.At last,we present the numerical experiments to verify the accuracy of the numerical algorithms and to show their long-time performance.The result shows that it is effective that we use Lie group methods to structure square-conservative schemes for the nonlinear Schr(?)dinger equation.