The smoothed wigner transform method: Precise coarse-scale simulation of wave propagation

The Wigner transform (WT) is a quadratic nonparametric phase-space density, introduced by Eugene Wigner for the phase-space formulation of quantum mechanics. It has since been used extensively in the asymptotic study of many wave propagation problems; it is well known however that it is not amenable to practical computations. The smoothed Wigner transform (SWT) is a variation of the WT, well known in the context of signal-processing and time-frequency analysis. Recently, it has been shown by the author that it can be used for the accurate and efficient numerical treatment of wave propagation problems, in contrast with the WT. More precisely, the SWT can be used to compute the solution at a chosen spatio-spectral resolution, i.e. leading to some degree of averaging. In this work, theoretical results are presented, which allow for a quantitative understanding of the errors introduced. A central idea, not common with more traditional techniques, is separating the averaging operator from the error, since we deal with an approximate coarse-scale solution. © 2007 American Institute of Physics.