This work outlines a time-domain numerical integration technique for linear hyperbolic partial differential equations sourced by distributions (Dirac $\delta$-functions and their derivatives). Such problems arise when studying binary black hole systems in the extreme mass ratio limit. We demonstrate that such source terms may be converted to effective domain-wide sources when discretized, and we introduce a class of time-steppers that directly account for these discontinuities in time integration. Moreover, our time-steppers are constructed to respect time reversal symmetry, a property that has been connected to conservation of physical quantities like energy and momentum in numerical simulations. To illustrate the utility of our method, we numerically study a distributionally-sourced wave equation that shares many features with the equations governing linear perturbations to black holes sourced by a point mass.

翻译：本文概述了一种针对分布源（狄拉克δ函数及其导数）的线性双曲型偏微分方程时域数值积分技术。此类问题出现在研究极端质量比极限下的双黑洞系统时。我们证明：当离散化后，此类源项可转化为等效全域源，并引入一类在时间积分中直接处理这些不连续性的时间步进器。此外，所构造的时间步进器满足时间反演对称性——该性质在数值模拟中与能量、动量等物理量的守恒密切相关。为验证方法的实用性，我们对一个分布源波动方程进行了数值研究，该方程与点质量源驱动的黑洞线性扰动方程具有许多共同特征。