Unconditionally stable discretization schemes of non-reflecting boundary conditions for the one-dimensional Schrِdinger equation

This paper addresses the problem of the construction of stable approximation schemes for the one-dimensional linear Schrödinger 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 developed 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 Dirichlet–Neumann (DN) and Neumann–Dirichlet (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 different schemes.