Efficient Structure-Preserving Algorithms For Two Kinds Of Nonlinear Schr(?)dinger Equations

With the development of science and technology,it has become one of the main research directions of numerical computing to construct an algorithm that preserves the original structure and is efficient for mathematical and physical models with special structures.This thesis will focus on constructing efficient structurepreserving algorithms for two kinds of nonlinear Schr(?)dinger equations.One is the Gross-Pitaevskii equation,which is a Hamiltonian system with not only conserved quantities such as mass and energy,but also symplectic and multi-symplectic structures.The other is the damped nonlinear Schr(?)dinger equation which is a conformal Hamiltonian system and has conformal symplectic,multi-symplectic structures,some other conformal conservation laws.Because these two types of equations are multidimensional nonlinear equations,the structure-preserving algorithms constructed for them are generally completely implicit and have low computational efficiency.To overcome this,this thesis will use the time splitting method to split the original equation into linear parts and nonlinear parts to solve.For the two-dimensional problems involved,the splitting method is also used to reduce the original problem from two-dimensional to one-dimensional,which can greatly improve the calculation efficiency.For the subproblem obtained by splitting,we will consider its structural characteristics and construct the corresponding structurepreserving algorithm,and then use the appropriate composition method to obtain an efficient algorithm.At the same time,some properties of the obtained algorithm,such as conservation law,stability,are proved theoretically.Finally,some numerical experiments are given to verify the accuracy of the constructed schemes,and verify that these schemes can stably calculate for a long time and preserve some conserved quantities.The structure of this thesis is arranged as follows:In Chapter 1,we briefly introduce some basic theories of structure-preserving algorithms for Hamiltonian systems and conformal Hamiltonian systems,and discuss the relevant properties and research status of the Gross-Pitaevskii equation and the damped nonlinear Schr(?)dinger equation at home and abroad.In Chapter 2,we study several commonly used methods for constructing efficient structure-preserving algorithms,mainly high order compact schemes for spatial discretization,time splitting methods,and so on.At the same time,the related knowledge of symplectic space is briefly studied,and several commonly used symplectic algorithms are given.In Chapter 3,we study the symplectic structure of the one-dimensional GrossPitaevskii equation,using a high-order compact scheme in space direction and a fourth-order precision symplectic scheme in time direction and splitting and combination method to discretize the equation,and obtain an efficient symplectic algorithm with fourth-order precision in the space-time direction.Subsequent theoretical analysis and numerical experiments also verify the effectiveness of the scheme.In Chapter 4,the multi-symplectic structure of the two-dimensional GrossPitaevskii equation with rotational angular momentum is mainly considered.We introduce the idea of local one-dimensional method,and use the high-order compact scheme to discretize the spatial derivative,and finally construct a high-order compact and local one-dimensional multi-symplectic scheme with second-order accuracy in time direction and fourth-order accuracy in space direction through appropriate combination Theoretical analysis and numerical experiments show that this scheme can maintain a certain structure and is very efficient.In Chapter 5,we mainly study the conformal multi-symplectic structure of twodimensional damped nonlinear Schr(?)dinger equation.The equation is decomposed into several one-dimensional conservation subsystems and a dissipative subsystem,and the subsystems are discretized or directly solved according to the characteristics of these systems.Finally,a high-order compact and local one-dimensional conformal multi-symplectic scheme is obtained through the combination method,which has second-order accuracy in the time direction and fourth-order accuracy in the space direction.The theoretical analysis and numerical experiments show that the scheme is stable,efficient and can maintain some structure of the original equation.In the Chapter 6,a summary for the whole thesis and a prospect for the future work are proposed.