SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page A3462-A3488, January 2020.
We propose a boundary method for the numerical simulation of time-dependent waves in three-dimensional (3D) exterior regions. The order of accuracy can be either second or fourth in both space and time. The method reduces a given initial boundary value problem for the wave equation to a set of operator equations at the boundary of the original domain. The reduction is based on a reformulation of the method of difference potentials. The resulting operator equations relate the solution and its normal derivative at the boundary. To solve these equations, one relies on the Huygens' principle. This yields an algorithm that works on a sliding time window of a finite nonincreasing duration. As a result, it allows one to avoid the ever increasing backward dependence of the solution on time. The major advantages of the proposed methodology are its reduced computational complexity (grid-independent on the boundary and sublinear in the volume), the capacity to handle curvilinear geometries using Cartesian finite difference time domain (FDTD) methods, and automatic and exact accounting for the far-field radiation conditions. In addition, the methodology facilitates solution of multiple similar problems al low individual cost per problem and guarantees uniform performance over arbitrarily long time intervals.

中文翻译：

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使用边界算子的3D外部非定常波传播问题的数值解

SIAM科学计算杂志，第42卷，第5期，第A3462-A3488页，2020年1月。
我们提出了一种边界方法，用于三维（3D）外部区域中随时间变化的波的数值模拟。在空间和时间上，精度的顺序可以是第二或第四。该方法将波动方程的给定初始边界值问题简化为原始域边界处的一组算子方程。减少是基于对电位差方法的重新设计。所得的算子方程将解及其边界处的正态导数相关。为了解决这些方程，我们依靠惠更斯原理。这产生了一种算法，该算法适用于持续时间有限的滑动时间窗口。结果，它可以避免解决方案对时间的日益增加的后向依赖性。所提出的方法的主要优点是：降低了计算复杂度（网格独立于体积中的边界和亚线性），具有使用笛卡尔有限差分时域（FDTD）方法处理曲线几何的能力，并且能够自动精确地计算远场辐射条件。此外，该方法有助于解决多个类似问题，同时降低每个问题的个人成本，并确保在任意长的时间间隔内保持一致的性能。