Communications in Mathematical Sciences

Volume 1 (2003)

Number 3

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

Pages: 501 – 556



Anton Arnold

Matthias Ehrhardt

Ivan Sofronov


We propose a way to efficiently treat the well-known transparent boundary conditions for the Schrödinger equation. Our approach is based on two ideas: to write out a discrete transparent boundary condition (DTBC) using the Crank-Nicolson finite difference scheme for the governing equation, and to approximate the discrete convolution kernel of DTBC by sum-of-exponentials for a rapid recursive calculation of the convolution.

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.


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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