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 $O(t_{ϵ}N\log N)$ per time step with N the number of spatial points and $t_{ϵ}$ the low rank for a prescribed accuracy requirement $ϵ>0$. Numerical experiments are presented to demonstrate the proposed method.

我们提出了一种有效的渐近格林函数方法，用于通过线性标量波动方程传播波。波首先被分成向前传播和向后传播的部分。随后，该方法结合惠更斯原理和几何光学近似，用于传播前向传播和后向传播波，其中基于惠更斯原理的格林函数积分用于传播波，并且格林函数由几何光学近似值近似。在通过短时泰勒级数展开获得几何光学近似中相位和幅度的解析近似后，推导了波的短时传播器，并且在适当的低秩近似之后，可以通过快速傅立叶变换有效地评估所得积分。短时传播器可以反复应用，以长时间传播波。为了将计算限制在感兴趣的有界域上，结合了具有复杂坐标拉伸变换的完美匹配层技术。该方法一阶准确，且具有复杂性$哦({吨}_{\epsilon }N\mathrm{日志}N)$每个时间步长与N空间点的数量和${吨}_{\epsilon }$ 规定精度要求的低等级 $\epsilon >0$. 数值实验被提出来证明所提出的方法。