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

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 Ω \Omega with artificial boundary conditions set on the arbitrarily shaped boundary of Ω \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.