A Wigner-measure analysis of the Dufort-Frankel scheme for the Schrödinger equation.

We apply the Wigner transform techniques to the analysis of the Dufort-Frankel difference scheme for the Schroedinger equation 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 the average value of observables when 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 whether caustics develop or not. Numerical tests are presented to help to interpret the theory and to compare the Dufort-Frankel scheme with other difference schemes for the Schrödinger equation.