Mathematical Models for Transport in Macroscopic and Mesoscopic Systems - Abstract

Ehrhardt, Matthias

Discrete transparent boundary conditions for the Schrödinger equation on circular domains

We propose transparent boundary conditions for the time-dependent Schrödinger equation on a circular computational domain. First we derive the two-dimensional discrete TBCs in conjunction with a conservative Crank-Nicolson-type finite difference scheme. The presented discrete boundary-valued problem is unconditionally stable and completely reflection-free at the boundary. Then, since the discrete TBCs for the Schrödinger equation with a spatially dependent potential include a convolution w.r.t. time with a weakly decaying kernel, we construct approximate discrete TBCs with a kernel having the form of a finite sum of exponentials, which can be efficiently evaluated by recursion. Finally, we describe several numerical examples illustrating the accuracy, stability and efficiency of the proposed method. We also comment briefly on the situation in different geometries, like straight line, wave guide and general convex geometry.