The stability and convergence of a difference scheme for the Schrödinger equation on an infinite domain by using artificial boundary conditions
ABSTRACT This paper is concerned with the numerical solution to the Schrödinger equation on an infinite domain. Two exact artificial boundary conditions are introduced to reduce the original problem into an initial boundary value problem with computational domain. Then, a fully discrete difference scheme is derived. The truncation errors are analyzed in detail. The unique solvability, stability and convergence with the convergence order of O(h3/2 + τ3/2h−1/2) are proved by the energy method. A numerical example is given to demonstrate the accuracy and efficiency of the proposed method. As a special case, the stability and convergence of the difference scheme proposed by Baskakov and Popov in 1991 is obtained.
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ABSTRACT: Bridges and Reich suggested the idea of multi-symplectic spectral discretization on Fourier space (4). Based on their theory, we investigate the multi-symplectic Fourier pseudospectral discretization of the nonlinear Schr¨ odinger equation (NLS) on real space. We show that the multi-symplectic semi-discretization of the nonlinear Schr¨ odinger equation with periodic boundary conditions has N (the number of the nodes) semi-discrete multi- symplectic conservation laws. The symplectic discretization in time of the semi-discretization leads to N full- discrete multi-symplectic conservation laws. We also prove a result relating to the spectral differentiation matrix. Numerical experiments are included to demonstrate the remarkable local conservation properties of multi-symplectic spectral discretizations.
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ABSTRACT: A finite-difference scheme is proposed for the one-dimensional time-dependent Schrödinger equation. We introduce an artificial boundary condition to reduce the original problem into an initial-boundary value problem in a finite-computational domain, and then construct a finite-difference scheme by the method of reduction of order to solve this reduced problem. This scheme has been proved to be uniquely solvable, unconditionally stable, and convergent. Some numerical examples are given to show the effectiveness of the scheme.Computers & Mathematics with Applications. 01/2005;