Communications in Mathematical Sciences

Volume 10 (2012)

Number 3

Discrete transparent boundary conditions for the Schrödinger equation on circular domains

Pages: 889 – 916



Anton Arnold (Institut für Analysis und Scientific Computing, Technische Universität Wien, Austria)

Matthias Ehrhardt (Lehrstuhl für Angewandte Mathematik und Numerische Analysis, Bergische Universität Wuppertal, Germany)

Maike Schulte (Institut für Numerische und Angewandte Mathematik, Universität Münster, Germany)

Ivan Sofronov (Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, Russia)


We propose transparent boundary conditions (TBCs) 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 finite difference scheme. The presented discrete initial boundary–value 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 with respect to 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. In numerical tests we finally illustrate the accuracy, stability, and efficiency of the proposed method.

As a by-product we also present a new formulation of discrete TBCs for the 1D Schrödinger equation, with convolution coefficients that have better decay properties than those from the literature.


two–dimensional Schrödinger equation, transparent boundary conditions, discrete convolution, sum of exponentials, Padé approximations, finite difference schemes

2010 Mathematics Subject Classification

35Q40, 45K05, 65M12

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