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Preprint 41-2007

Discrete transparent boundary conditions for the Schrödinger equation on circular domains

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

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 41-2007

MSC 2000

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

Abstract :
We propose transparent boundary conditions for the time-dependent Schrödinger equation on a circular computational domain. First we derive the two-dimensional discrete TBCs in conjunction with a conservative Crank-Nicolson-type finite difference scheme. The presented discrete boundary-valued problem is unconditionally stable and completely reflection-free at the boundary. Then, since the discrete TBCs for the Schrödinger equation with a spatially dependent potential include a convolution w.r.t. time with a weakly decaying kernel, we construct approximate discrete TBCs with a kernel having the form of a finite sum of exponentials, which can be efficiently evaluated by recursion. Finally, we describe several numerical examples illustrating the accuracy, stability and efficiency of the proposed method.

Keywords : two-dimensional 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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