Mathematics of Computation

Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions

Author(s): Xavier Antoine; Christophe Besse; Vincent Mouysset.
Journal: Math. Comp. 73 (2004), 1779-1799.
MSC (2000): Primary 65M12, 35Q40, 58J40, 26A33, 58J47
Posted: January 23, 2004
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Abstract: This paper adresses the construction and study of a Crank-Nicolson-type discretization of the two-dimensional linear Schrödinger equation in a bounded domain $\Omega$ with artificial boundary conditions set on the arbitrarily shaped boundary of $\Omega$ . These conditions present the features of being differential in space and nonlocal in time since their definition involves some time fractional operators. After having proved the well-posedness of the continuous truncated initial boundary value problem, a semi-discrete Crank-Nicolson-type scheme for the bounded problem is introduced and its stability is provided. Next, the full discretization is realized by way of a standard finite-element method to preserve the stability of the scheme. Some numerical simulations are given to illustrate the effectiveness and flexibility of the method.

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Xavier Antoine
Affiliation: Laboratoire de Mathématiques pour l'Industrie et la Physique, CNRS UMR 5640, UFR MIG, 118, route de Narbonne, 31062 Toulouse Cedex 4, France
Email: antoine@mip.ups-tlse.fr

Christophe Besse
Affiliation: Laboratoire de Mathématiques pour l'Industrie et la Physique, CNRS UMR 5640, UFR MIG, 118, route de Narbonne, 31062 Toulouse Cedex 4, France
Email: besse@mip.ups-tlse.fr

Vincent Mouysset
Affiliation: Laboratoire de Mathématiques pour l'Industrie et la Physique, CNRS UMR 5640, UFR MIG, 118, route de Narbonne, 31062 Toulouse Cedex 4, France
Email: mouysset@mip.ups-tlse.fr

DOI: 10.1090/S0025-5718-04-01631-X
PII: S 0025-5718(04)01631-X
Keywords: Schr\"{o}dinger equation, non-reflecting boundary condition, stability, semi-discrete Crank-Nicolson-type scheme, finite-element methods
Received by editor(s): November 7, 2002
Received by editor(s) in revised form: April 7, 2003
Posted: January 23, 2004
Copyright of article: Copyright 2004, American Mathematical Society

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