An asymptotic Green's function method for the wave equation

We present an effective asymptotic Green's function method for propagating the waves through the linear scalar wave equation. The wave is first split into its forward-propagating and backward-propagating parts. Following that, the method, which combines the Huygens' principle and the geometrical optics approximations, is designed to propagate the forward-propagating and backward-propagating waves, where an integral with Green's functions that is based on the Huygens' principle is used to propagate the waves, and the Green's functions are approximated by the geometrical optics approximations. 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 by fast Fourier transform after appropriate lowrank approximations. 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 is incorporated. The method is first-order accurate, and has complexity O(tϵNlog⁡N) per time step with N the number of spatial points and tϵ the low rank for a prescribed accuracy requirement ϵ>0. Numerical experiments are presented to demonstrate the proposed method.