SIAM Journal on Scientific Computing, Volume 42, Issue 2, Page A1348-A1379, January 2020.
This paper proposes a frequency/time hybrid integral-equation method for the time-dependent wave equation in two- and three-dimensional spatial domains. Relying on Fourier transformation in time, the method utilizes a fixed (time-independent) number of frequency-domain integral-equation solutions to evaluate, with superalgebraically small errors, time-domain solutions for arbitrarily long times. The approach relies on two main elements, namely: (1) a smooth time-windowing methodology that enables accurate band-limited representations for arbitrarily long time signals and (2) a novel Fourier transform approach which, in a time-parallel manner and without causing spurious periodicity effects, delivers numerically dispersionless spectrally accurate solutions. A similar hybrid technique can be obtained on the basis of Laplace transforms instead of Fourier transforms, but we do not consider the Laplace-based method in the present contribution. The algorithm can handle dispersive media, it can tackle complex physical structures, it enables parallelization in time in a straightforward manner, and it allows for time leaping---that is, solution sampling at any given time $T$ at $\mathcal{O}(1)$-bounded sampling cost, for arbitrarily large values of $T$, and without requirement of evaluation of the solution at intermediate times. The proposed frequency-time hybridization strategy, which generalizes to any linear partial differential equation in the time domain for which frequency-domain solutions can be obtained (including, e.g., the time-domain Maxwell equations) and which is applicable in a wide range of scientific and engineering contexts, provides significant advantages over other available alternatives, such as volumetric discretization, time-domain integral equations, and convolution quadrature approaches.

中文翻译：

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高阶，无色散的“快速混合”波动方程求解器。第一部分：通过事件字段窗口化和重新中心化的O（1）采样成本

SIAM科学计算杂志，第42卷，第2期，第A1348-A1379页，2020年1月。
针对二维和三维空间域中随时间变化的波动方程，提出了一种频率/时间混合积分方程方法。该方法依靠时间上的傅立叶变换，利用固定数量（与时间无关）的频域积分方程解，以极小的代数误差评估任意长时间的时域解。该方法依赖于两个主要元素，即：（1）一种平滑的时间窗口方法，该方法可对任意长时间信号进行精确的带限表示；（2）一种新颖的傅立叶变换方法，该方法以时间并行的方式且没有引起寄生周期效应，提供数值上无色散的光谱精确解。可以基于拉普拉斯变换而不是傅立叶变换来获得类似的混合技术，但是在本文中我们不考虑基于拉普拉斯的方法。该算法可以处理分散的介质，可以处理复杂的物理结构，可以以直接的方式实现时间并行化，并且可以实现时间跳跃-也就是说，在任何给定的时间$ T $处的解决方案采样为$ \ mathcal { O}（1）$限制了采样成本，对于$ T $的任意大值，并且不需要在中间时间评估解决方案。拟议的频率-时间杂交策略，可以将其推广到时域中可以获取频域解的任何线性偏微分方程（例如，