Two fundamental difficulties are encountered in the numerical evaluation of time-dependent layer potentials. One is the quadratic cost of history dependence, which has been successfully addressed by splitting the potentials into two parts—a local part that contains the most recent contributions and a history part that contains the contributions from all earlier times. The history part is smooth, easily discretized using high-order quadratures, and straightforward to compute using a variety of fast algorithms. The local part, however, involves complicated singularities in the underlying Green’s function. Existing methods, based on exchanging the order of integration in space and time, are able to achieve high-order accuracy, but are limited to the case of stationary boundaries. Here, we present a new quadrature method that leaves the order of integration unchanged, making use of a change of variables that converts the singular integrals with respect to time into smooth ones. We have also derived asymptotic formulas for the local part that lead to fast and accurate hybrid schemes, extending earlier work for scalar heat potentials and applicable to moving boundaries. The performance of the overall scheme is demonstrated via numerical examples.

中文翻译：

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关于二维运动几何中非定常斯托克斯层势的准确评估

在时间相关层电势的数值评估中遇到两个基本困难。一种是历史依赖的二次成本，已成功地解决了这一问题，方法是将潜力分成两部分-包含最新贡献的本地部分和包含所有较早时期的贡献的历史部分。历史记录部分平滑，易于使用高阶正交离散化，并且可以使用多种快速算法直接进行计算。但是，局部部分在基础格林函数中涉及复杂的奇点。基于交换时空积分顺序的现有方法能够实现高阶精度，但仅限于固定边界的情况。这里，我们提出了一种新的正交方法，该方法不改变积分顺序，而是利用变量的变化将相对于时间的奇异积分转换为平滑积分。我们还导出了局部曲线的渐近公式，这些公式导致了快速，准确的混合方案，扩展了标量热势的早期工作，并适用于移动边界。通过数值实例证明了整体方案的性能。