Summary: Unconditionally stable discretization schemes of
non-reflecting boundary conditions for the one-dimensional
Schroodinger equation
X. Antoine, C. Besse *
Laboratoire de Matheematiques pour lÕIndustrie et la Physique, UMR 5640, Complexe Scientifique de Rangueil,
31077 Toulouse cedex 4, France
Received 28 March 2002; received in revised form 21 October 2002; accepted 21 February 2003
Abstract
This paper addresses the problem of the construction of stable approximation schemes for the one-dimensional
linear Schroodinger equation set in an unbounded domain. After a study of the initial boundary-value problem in a
bounded domain with a transparent boundary condition, some unconditionally stable discretization schemes are de-
veloped for this kind of problem. The main difficulty is linked to the involvement of a fractional integral operator
defining the transparent operator. The proposed semi-discretization of this operator yields with a very different point of
view the one proposed by Yevick, Friese and Schmidt [J. Comput. Phys. 168 (2001) 433]. Two possible choices of
transparent boundary conditions based on the Dirichlet­Neumann (DN) and Neumann­Dirichlet (ND) operators are
presented. To preserve the stability of the fully discrete scheme, conform Galerkin finite element methods are employed
for the spatial discretization. Finally, some numerical tests are performed to study the respective accuracy of the dif-
ferent schemes.
Ó 2003 Elsevier Science B.V. All rights reserved.
1. Introduction

Source: Antoine, Xavier - Institut de Mathématiques Élie Cartan, Université Henri Poincaré - Nancy 1

Collections: Mathematics