Computational High-Frequency Quantum Wave Propagation by the Level Set Method

Dr. Hailiang Liu

Iowa State University

Abstract:  Numerical simulation of high frequency wave propagation is important in many applications. We recently develop a new level set method for multi-phase computations for linear Schodinger equations with efficiently highly oscillating initial data. The high-frequency asymptotics of these equations lead to the well known WKB system, where the phase evolves according to a nonlinear Hamilton-Jacobi equation. For the Hamilton-Jacobi equation we show how the wave fronts are constructed, and the multi-valued phases are realized via intersection of several zero level sets in an extended phase space. We shall also outline how this idea can be used for the computation of multivalued solutions to a general class of nonlinear first-order equations in arbitrary space dimension. The level set approach we use automatically handles the multi-valued solutions that appear.

This talk reflects recent joint investigation with Li-Tien Cheng (UCSD) and Stanley Osher (UCLA).