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Preprint 753-2002

Discrete transparent boundary conditions for the Schrödinger equation: Fast calculation, approximation, and stability

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Author(s) : Anton Arnold , Matthias Ehrhardt , Ivan Sofronov

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 753-2002

MSC 2000

65M12 Stability and convergence of numerical methods
35Q40 Equations from quantum mechanics
45K05 Integro-partial differential equations

Abstract :
This paper is concerned with transparent boundary conditions (TBCs) for the time-dependent Schrödinger equation in one and two dimensions. Discrete TBCs are introduced in the numerical simulations of whole space problems in order to reduce the computational domain to a finite region. Since the discrete TBC for the Schrödinger equation includes a convolution w.r.t. time with a weakly decaying kernel, its numerical evaluation becomes very costly for large-time simulations.

As a remedy we construct approximate TBCs with a kernel having the form of a finite sum-of-exponentials, which can be evaluated in a very efficient recursion. We prove stability of the resulting initial-boundary value scheme, give error estimates for the considered approximation of the boundary condition, and illustrate the efficiency of the proposed method on several examples.

Keywords : Schrödinger equation, transparent boundary conditions, discrete convolution, sum of exponentials, Padé approximations, finite difference schemes

Notes :
(submitted to: "Communications in Mathematical Sciences")

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