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：本文提出了一种针对由分布（狄拉克$\delta$函数及其导数）驱动的线性双曲型偏微分方程的时域数值积分技术。这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，此类源项在离散化后可转化为有效的全域源，并引入一类直接处理时间积分中这些不连续性的时间步进器。此外，我们的时间步进器设计遵循时间反演对称性，这一性质与数值模拟中能量、动量等物理量的守恒密切相关。为验证该方法的实用性，我们数值研究了与点质量驱动黑洞线性扰动方程具有许多共同特征的分布源波动方程。