A Wigner-Measure Analysis of the Dufort--Frankel Scheme for the Schrödinger Equation

Abstract

We apply Wigner transform techniques to the analysis of the Dufort--Frankel difference scheme for the Schrödinger equation and to the continuous analogue of the scheme in the case of a small (scaled) Planck constant (semiclassical regime). In this way we are able to obtain sharp conditions on the spatial-temporal grid which guarantee convergence for average values of observables as the Planck constant tends to zero. The theory developed in this paper is not based on local and global error estimates and does not depend on whether or not caustics develop. Numerical test examples are presented to help interpret the theory and to compare the Dufort--Frankel scheme to other difference schemes for the Schr{ödinger equation.