Generalization of the Weyl-Moyal calculus and computational homogenization of wave propagation
given at strobl07 (22.06.07 11:30)
In the study of wave propagation over distances much longer than the typical wavelength phase-space methods are often used, i.e. spectral densities are used as a homogenized representation of the wavefield, and kinetic equations are constructed for their evolution.
The Wigner Transform (WT) is a nonlinear, nonparametric spectral density, first introduced by E. Wigner in the context of quantum mechanics. Recently it has been extensively used in the formulation of phase-space models for a variety of problems, such as geometrical optics limits, periodic problems, nonlinear and/or random waves. Physical areas of application include semiconductors, linear and nonlinear optics, water waves and more.
However, the WT features counterintuitive ‘interference components’, which make computation and interpretation problematic. To face this, variants such as Wigner measures are typically used in practice.
We use smoothed Wigner Transforms (SWT) to study wave propagation. We present the derivation of new, to the best of our knowledge, exact equations of motion for the SWT covering a broad class of wave propagation problems. As a special case we get exact equations of motion for spectrograms of wave fields. These equations are typically pseudodifferential.
The new equations are used for the construction of a reliable, efficient, homogenized (slow-scale) numerical solver for the Schrodinger equation. The new solver successfully captures the spatial structure of caustics, while clearly outperforming conventional, full (i.e. not homogenized) solvers. A subtle issue which we discuss is the construction of an appropriate `slow-scale error', i.e. a quantitative measure of how good an approximation our SWT solution offers; exact solutions are used to test our 'slow-scale error', with good results.