The stability and convergence of a difference scheme for the Schrödinger equation on an infinite domain by using artificial boundary conditions

Department of Mathematics, Hong Kong Baptist University, Chiu-lung, Kowloon City, Hong Kong
Journal of Computational Physics (Impact Factor: 2.43). 05/2006; 214(1):209-223. DOI: 10.1016/


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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    • "For more literature works that focus on the numerical treatment of (2.3) and (2.4), we shall refer to [4] [27] [29] [34] [35], etc. To construct local ABCs, the boundary should be almost transparent for a plane wave of the form [30] "
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    ABSTRACT: The paper is concerned with the numerical solution of Schro ̈dinger equa- tions on an unbounded spatial domain. High-order absorbing boundary conditions for one-dimensional domain are derived, and the stability of the reduced initial bound- ary value problem in the computational interval is proved by energy estimate. Then a second order finite difference scheme is proposed, and the convergence of the scheme is established as well. Finally, numerical examples are reported to confirm our error estimates of the numerical methods.
    Full-text · Article · Jun 2011 · Communications in Computational Physics
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    • "For linear problems , many strategies have been developed to obtain accurate and efficient boundary conditions, such as [7] [8] for hyperbolic wave equations, [9] [10] for elliptic equations, and [11] [12] for parabolic equations. In the case of the linear Schrödinger equation, there are also several works [13] [14] [15] [16] [17] [18] [19] developing transparent boundary conditions and studying their difference approximations and stability. They utilized the integral transform (Laplace or Fourier transform) or series expansion method to construct accurate boundary conditions which are in nonlocal forms. "
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    ABSTRACT: We propose an adaptive approach in picking the wave-number parameter of absorbing boundary conditions for Schrödinger-type equations. Based on the Gabor transform which captures local frequency information in the vicinity of artificial boundaries, the parameter is determined by an energy-weighted method and yields a quasi-optimal absorbing boundary conditions. It is shown that this approach can minimize reflected waves even when the wave function is composed of waves with different group velocities. We also extend the split local absorbing boundary (SLAB) method [Z. Xu, H. Han, Phys. Rev. E 74 (2006) 037704] to problems in multi-dimensional nonlinear cases by coupling the adaptive approach. Numerical examples of nonlinear Schrödinger equations in one and two dimensions are presented to demonstrate the properties of the discussed absorbing boundary conditions.
    Preview · Article · Oct 2006 · Journal of Computational Physics
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    ABSTRACT: The numerical simulation of the solution to a modified KdV equation on the whole real axis is considered in this paper. Based on the work of Fokas (Comm Pure Appl Math 58(5):639–670, 2005), a kind of exact nonreflecting boundary conditions which are suitable for numerical purposes are presented with the inverse scattering theory. With these boundary conditions imposed on the artificially introduced boundary points, a reduced problem defined on a finite computational interval is formulated. The discretization of the nonreflecting boundary conditions is studied in detail, and a dual-Petrov–Galerkin spectral method is proposed for the numerical solution to the reduced problem. Some numerical tests are given, which validate the effectiveness, and suggest the stability of the proposed scheme.
    No preview · Article · Nov 2006 · Numerische Mathematik
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