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

DiscoTEX 1.0: Discontinuous collocation and implicit-turned-explicit  (IMTEX) integration symplectic, symmetric numerical algorithms with higher  order jumps for differential equations I: numerical black hole perturbation  theory applications