Communications in Mathematical Sciences

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

Anton Arnold, Matthias Ehrhardt, and Ivan Sofronov

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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.

Article information

Source
Commun. Math. Sci., Volume 1, Number 3 (2003), 501-556.

Dates
First available in Project Euclid: 21 August 2009

Permanent link to this document
https://projecteuclid.org/euclid.cms/1250880098

Mathematical Reviews number (MathSciNet)
MR2069942

Zentralblatt MATH identifier
1085.65513

Subjects
Primary: 65M12: Stability and convergence of numerical methods 35Q40: PDEs in connection with quantum mechanics 45K05: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20]

Keywords
Schrödinger equation transparent boundary conditions discrete convolution sum of exponentials Padé approximations finite difference schemes

Citation

Arnold, Anton; Ehrhardt, Matthias; Sofronov, Ivan. Discrete transparent boundary conditions for the Schrödinger equation: fast calculation, approximation, and stability. Commun. Math. Sci. 1 (2003), no. 3, 501--556. https://projecteuclid.org/euclid.cms/1250880098


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