Artificial Boundary Conditions for Schrödinger-type Equations and their Numerical Approximation

Abstract. The construction of a hierarchy of high-frequency microlocal artificial boundary conditions for the two-dimensional linear Schrödinger equation is proposed. These conditions are derived for a circular boundary and are next extended to a general arbitrarily-shaped boundary. They present the features of being differential in space and non-local in time since their definition involves some temporal fractional derivative operators. The well-posedness of the continuous truncated initial boundary value problem is provided. A semi-discrete Crank-Nicolson-type scheme for the bounded problem is introduced and a stability result is given. Next, the full discretization is realized by the 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.