Dirac $\delta-$ distributionally sourced differential equations emerge in many dynamical physical systems from machine learning, finance, neuroscience, and seismology to black hole perturbation theory. These systems lack exact analytical solutions and are thus best tackled numerically. We describe a generic numerical algorithm which constructs discontinuous spatial and temporal discretisations by operating on discontinuous Lagrange and Hermite interpolation formulae, respectively. By solving the distributionally sourced wave equation, possessing analytical solutions, we demonstrate that numerical weak-form solutions can be recovered to high-order accuracy by solving a first-order reduced system of ODEs. The method-of-lines framework is applied to the \texttt{DiscoTEX} algorithm i.e. through \underline{dis}continuous \underline{co}llocation with implicit\underline{-turned-explicit} integration methods which are symmetric and conserve symplectic structure. Furthermore, the main application of the algorithm is proved by calculating the amplitude at any desired location within the numerical grid, including at the position (and at its right and left limit) where the wave- (or wave-like) equation is discontinuous via interpolation using \texttt{DiscoTEX}. This is demonstrated, firstly by solving the wave- (or wave-like) equation and comparing the numerical weak-form solution to the exact solution. We further demonstrate how to reconstruct the gravitational metric perturbations from weak-form numerical solutions of a non-rotating black hole, which do not have known exact analytical solutions, and compare them against state-of-the-art frequency domain results. We conclude by motivating how \texttt{DiscoTEX}, and related numerical algorithms, both open a promising new alternative waveform generation route for modelling highly asymmetric binaries and complement current frequency domain methods.

翻译：狄拉克$\delta-$分布源微分方程出现在从机器学习、金融学、神经科学、地震学到黑洞微扰理论等诸多动态物理系统中。这些系统缺乏精确的解析解，因此最适合用数值方法处理。我们描述了一种通用数值算法，该算法分别通过对间断拉格朗日和埃尔米特插值公式进行操作，构建了间断的空间与时间离散格式。通过求解具有解析解的分布源波动方程，我们证明了通过求解一阶约化常微分方程组，可以高阶精度地恢复数值弱形式解。将线法框架应用于\texttt{DiscoTEX}算法，即通过具有对称性且保持辛结构的隐式转显式积分方法进行间断配置。此外，通过使用\texttt{DiscoTEX}进行插值，计算数值网格内任意期望位置（包括波动（或类波动）方程间断处的位置及其左右极限）的振幅，验证了该算法的主要应用。我们首先通过求解波动（或类波动）方程，并将数值弱形式解与精确解进行比较来证明这一点。我们进一步演示了如何从无旋转黑洞的弱形式数值解重构引力度规微扰（这些解没有已知的精确解析解），并将其与最先进的频域结果进行对比。最后，我们阐述了\texttt{DiscoTEX}及相关数值算法如何为高度不对称双星系统的建模开辟一条有前景的新波形生成途径，并对当前频域方法形成补充。