Weak Ill-Posedness of Spatial Discretizations of Absorbing Boundary Conditions for Schrödinger-Type Equations

When we wish to solve numerically a differential problem defined on an infinite domain, it is necessary to consider a finite subdomain and to use artificial boundary conditions in such a way that the solutions in the finite subdomain approximate the original solution. These boundary conditions are called absorbing when small reflections to the interior domain are allowed. In this paper, we develop a general class of absorbing boundary conditions for Schrödinger-type equations by using rational approximations to the transparent boundary conditions. With this approach, previous absorbing boundary conditions in the literature are included in this class. We use the method of lines for the discretization of the initial boundary value problems obtained this way. We show that the ordinary differential systems that arise after the spatial discretization are weakly ill-posed, explaining a previous conjecture of Fevens and Jiang. The time discretization is carried out with A-stable Runge--Kutta methods, where the high order ones may be used to compensate for the possible troubles present in the problems semidiscretized in space.