Contents Online
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
DOI: https://dx.doi.org/10.4310/CMS.2003.v1.n3.a7
Authors
Abstract
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.
Keywords
Schrödinger equation, transparent boundary conditions, discrete convolution, sum of exponentials, Padé approximations, finite difference schemes
2010 Mathematics Subject Classification
35Q40, 45K05, 65M12
Published 1 January 2003