Pa. Markowich et al., Numerical approximation of quadratic observables of Schrodinger-type equations in the semi-classical limit, NUMER MATH, 81(4), 1999, pp. 595-630
We apply Wigner-transform techniques to the analysis of difference methods
for Schrodinger-type equations in the case of a small Planck constant. In t
his way we are able to obtain sharp conditions on the spatial-temporal grid
which guarantee convergence for average values of observables as the Planc
k constant tends to zero. The theory developed in this paper is not based o
n local and global error estimates and does not depend on whether caustics
develop or not. Numerical test examples are presented to help interpret the
theory. Mathematics Subject Classification (1991): 65M12.