Kinetic Description of Multiscale Phenomena

The Annual Kinetic FRG Meeting
September 21-25, 2009

CSIC Building (#406), Seminar Room 4122.

Global recovery of high frequency wave fields

Dr. Hailiang Liu

Iowa State University

Abstract:  Computation of high frequency solutions to wave equations is important in many applications, and notoriously difficult in resolving wave oscillations. Gaussian beams are asymptotically valid high frequency solutions concentrated on a single curve through the physical domain, and superposition of Gaussian beams provides a powerful tool to generate more general high frequency solutions to PDEs. An alternative way to compute Gaussian beam components such as phase, amplitude and Hessian of the phase, is to capture them in phase space by solving Liouville type equations on uniform grids. In this talk I shall present a systematic construction of asymptotic high frequency wave fields from computations in phase space for acoustic wave equations (also for the Schrödinger equation). The k-th order Gaussian beam superposition is shown to converge to the original wave field in the energy norm, at an optimal rate in arbitrary spatial dimension.