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

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.