Discrete transparent boundary conditions for the two dimensional Schrödinger equation
Please always quote using this URN:urn:nbn:de:0296-matheon-3411
- This paper is concerned with transparent boundary conditions (TBCs) for the time-dependent Schrödinger equation on a circular domain. Discrete TBCs are introduced in the numerical simulations of problems on unbounded domains in order to reduce the computational domain to a finite region in order to make this problem feasible for numerical simulations. The main focus of this article is on the appropriate discretization of such TBCs for the two-dimensional Schrödinger equation in conjunction with a conservative Crank-Nicolson-type finite difference discretization. The presented discrete TBCs yield an unconditionally stable numerical scheme and are completely reflection-free at the boundary. Furthermore we prove concisely the stability of the recurrence formulas used to obtain the convolution coefficients of the new discrete TBC for a spatially dependent potential.
Author: | Anton Arnold, Matthias Ehrhardt, Maike Schulte, Ivan Sofronov |
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URN: | urn:nbn:de:0296-matheon-3411 |
Referee: | Volker Mehrmann |
Document Type: | Preprint, Research Center Matheon |
Language: | English |
Date of first Publication: | 2006/05/31 |
Release Date: | 2006/05/31 |
Tag: | |
Institute: | Technische Universität Berlin |
MSC-Classification: | 35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Qxx Equations of mathematical physics and other areas of application [See also 35J05, 35J10, 35K05, 35L05] / 35Q40 PDEs in connection with quantum mechanics |
45-XX INTEGRAL EQUATIONS / 45Kxx Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20] / 45K05 Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20] | |
65-XX NUMERICAL ANALYSIS / 65Mxx Partial differential equations, initial value and time-dependent initial- boundary value problems / 65M12 Stability and convergence of numerical methods | |
Preprint Number: | 332 |