 
Summary: MATHEMATICS OF COMPUTATION
Volume 73, Number 248, Pages 17791799
S 00255718(04)01631X
Article electronically published on January 23, 2004
NUMERICAL SCHEMES FOR THE SIMULATION
OF THE TWODIMENSIONAL SCHR¨ODINGER EQUATION
USING NONREFLECTING BOUNDARY CONDITIONS
XAVIER ANTOINE, CHRISTOPHE BESSE, AND VINCENT MOUYSSET
Abstract. This paper adresses the construction and study of a CrankNicol
sontype discretization of the twodimensional linear Schr¨odinger equation in
a bounded domain with artificial boundary conditions set on the arbitrarily
shaped boundary of . These conditions present the features of being differ
ential in space and nonlocal in time since their definition involves some time
fractional operators. After having proved the wellposedness of the continuous
truncated initial boundary value problem, a semidiscrete CrankNicolsontype
scheme for the bounded problem is introduced and its stability is provided.
Next, the full discretization is realized by way of a standard finiteelement
method to preserve the stability of the scheme. Some numerical simulations
are given to illustrate the effectiveness and flexibility of the method.
1. Introduction
