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

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.