Dirac delta distributionally sourced differential equations emerge in many dynamical physical systems from neuroscience to black hole perturbation theory. Most of these lack exact analytical solutions and are thus best tackled numerically. This work describes a generic numerical algorithm which constructs discontinuous spatial and temporal discretisations by operating on discontinuous Lagrange and Hermite interpolation formulae recovering higher order accuracy. It is shown by solving the distributionally sourced wave equation, which has analytical solutions, that numerical weak-form solutions can be recovered to high order accuracy by solving a first-order reduced system of ordinary differential equations. The method-of-lines framework is applied to the DiscoTEX algorithm i.e through discontinuous collocation with implicit-turned-explicit (IMTEX) integration methods which are symmetric and conserve symplectic structure. Furthermore, the main application of the algorithm is proved, for the first-time, 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 DiscoTEX. This is shown, firstly by solving the wave- (or wave-like) equation and comparing the numerical weak-form solution to the exact solution. Finally, one shows how to reconstruct the scalar and gravitational metric perturbations from weak-form numerical solutions of a non-rotating black hole, which do not have known exact analytical solutions, and compare against state-of-the-art frequency domain results. One concludes by motivating how DiscoTEX, and related algorithms, open a promising new alternative Extreme-Mass-Ratio-Inspiral (EMRI)s waveform generation route via a self-consistent evolution for the gravitational self-force programme in the time-domain.

翻译：狄拉克δ分布源微分方程广泛存在于从神经科学到黑洞微扰理论等众多动态物理系统中。此类方程大多缺乏精确解析解，因此适合采用数值方法处理。本文提出一种通用数值算法，通过作用于不连续拉格朗日与埃尔米特插值公式，构建不连续空间与时间离散格式，实现高阶精度恢复。通过求解具有解析解的分布源波动方程，本文证明：通过求解一阶约化常微分方程组，可恢复高阶精度的数值弱形式解。将方法线框架应用于DiscoTEX算法——即基于不连续配点与隐式转显式（IMTEX）积分方法，该方法具有对称性并保持辛结构。进一步，本文首次验证了该算法的主要应用：通过DiscoTEX插值，可计算数值网格内任意指定位置（包括波动（或类波动）方程不连续点处及其左右极限）的振幅。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比较予以验证。最后，本文展示了如何从不具有已知精确解析解的非旋转黑洞弱形式数值解中重构标量度规与引力度规扰动，并与当前最先进的频域结果进行对比。结论部分阐述了DiscoTEX及相关算法如何通过时域引力自洽演化程序，为极端质量比旋近（EMRI）波形生成开辟一条有前景的新路径。