  The numerical simulation of wave propagation in semiclassical
(high-frequency) problems is well known to pose a formidable challenge. In this
work, a new phase-space approach for the numerical simulation of semiclassical
wave propagation, making use of the smoothed Wigner Transform (SWT), is
proposed. There are numerous works which use the Wigner Transform (WT) in the
study of a variety of wave propagation problems including high-frequency limits
for linear, nonlinear and/or random waves. The WT however is well known to
present significant difficulties in the formulation of numerical schemes.
Working with concrete examples for the semiclassical linear Schrodinger
equation it is seen that the SWT approach is indeed significantly faster (in a
well-defined sense) to work with than the WT and than full numerical solutions
of the original equation in the semiclassical regime. Comparisons with exact
and numerical solutions are used to keep track of numerical errors.
