Multigrid Methods For Solving Schr?dinger Equation

Schr?dinger equations are widely used in different fields,such as nuclear physics,optics,life sciences,etc.However,the complex problems in practice are not easy to obtain their exact solutions.With the development of computer technology,the numerical solution of these complex problems is paid more and more attention.Nevertheless,due to the special coupling of Schr?dinger equations,which involves complex functions or even nonlinearity,it is still difficult to use classical discrete methods in large-scale calculation and longtime simulation.So it is necessary to construct an efficient numerical solution.This paper studies the fast methods of Schr?dinger equations,which is mainly divided into two parts.The first part,aiming at the initial boundary value problem of nonlinear Schr?dinger equation,studies its two-grid method.The concrete work is as follows:using the improved Crank-Nicolson scheme and linear Galerkin finite element method to discretize the problem,it is theoretically proved that the fully discretized scheme is mass conserved,and the L~2 and H~1 norm are estimated by error splitting technique.Then,two mesh algorithms are constructed for the above discrete systems.That is,on the coarse grid,we solve a coupled algebraic system.On the fine grid,we use the known quantities of the corresponding time layer of the previous time layer and the coarse grid to make a certain combination to obtain two decoupling systems,and give the convergence analysis of the first algorithm.Finally,numerical experiments show that the new two-grid algorithm can achieve the optimal convergence order in the H~1 norm,which verifies the efficiency of the algorithm.The second part,aiming at the initial boundary value problem of the linear Schr?dinger equation,constructs the CascadicMultigrid Method and the Extrapolation Cascadic Multigrid Method.Through numerical experiments,the new algorithm is comparedwith the direct conjugate gradient method,and the numerical results show that the Extrapolation Cascadic Multigrid Method has better convergence and efficiency.