Analysis, Dynamics and Applications Seminar

Green’s Function Method for Wave Equations

An effective asymptotic Green’s function method for propagating the waves through the linear wave equation is first presented. First, the wave is split in its forward and backward propagating parts.  Then, the method, which combines Huygens’ principle and geometrical optics approximations, is designed to propagate the forward and backward propagating waves, where an integral with the Green’s function that is based on Huygens’ principle is used to propagate the waves. Upon obtaining analytic approximations for the phase and amplitude in the geometrical optics approximations through short-time Taylor series expansions, a short-time propagator for the waves is derived and the resulting integral can be evaluated efficiently via Fast Fourier Transforms, after an appropriate lowrank approximation. The short-time propagator can be applied repeatedly to propagate the waves for a long time. In order to restrict the computation onto a bounded domain of interest, the perfectly matched layer technique with complex coordinate stretching transformation in incorporated into the short-time propagator. Numerical experiments are then presented and discussed.  After, a short-time propagator is in development for a vector wave equation. The derivation and implementation is presented in a similar fashion as the scalar wave equation. Preliminary results will be presented.

Speaker will be in-person.