WIAS Preprint No. 1344, (2008)

Discrete transparent boundary conditions for the Schrödinger equation on circular domains


  • Arnold, Anton
  • Ehrhardt, Matthias
  • Schulte, Maike
  • Sofronov, Ivan

2010 Mathematics Subject Classification

  • 65M12 35Q40 45K05


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


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 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. 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.

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