Summary: DISCRETE TRANSPARENT BOUNDARY CONDITIONS FOR THE
SCHR¨ODINGER EQUATION ON CIRCULAR DOMAINS
ANTON ARNOLD , MATTHIAS EHRHARDT , MAIKE SCHULTE §, AND IVAN
SOFRONOV ¶
Abstract. We propose transparent boundary conditions (TBCs) for the time­dependent
Schr¨odinger equation on a circular computational domain. First we derive the two­dimensional
discrete TBCs in conjunction with a conservative Crank­Nicolson finite difference scheme. The pre-
sented discrete initial boundary­value problem is unconditionally stable and completely reflection­
free at the boundary. Then, since the discrete TBCs for the Schr¨odinger 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. In numerical tests we finally illustrate the accuracy, stability,
and efficiency of the proposed method.
As a by-product we also present a new formulation of discrete TBCs for the 1D Schr¨odinger
equation, with convolution coefficients that have better decay properties than those from the litera-
ture.
Key words. two­dimensional Schr¨odinger equation, transparent boundary conditions, discrete
convolution, sum of exponentials, Pad´e approximations, finite difference schemes
subject classifications. 65M12, 35Q40, 45K05
1. Introduction

Source: Arnold, Anton - Institut für Analysis und Scientific Computing, Technische Universität Wien

Collections: Mathematics