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Dirac delta distributionally sourced differential equations emerge in manydynamical physical systems from neuroscience to black hole perturbation theory.Most of these lack exact analytical solutions and are thus best tacklednumerically. This work describes a generic numerical algorithm which constructsdiscontinuous spatial and temporal discretisations by operating ondiscontinuous Lagrange and Hermite interpolation formulae recovering higherorder accuracy. It is shown by solving the distributionally sourced waveequation, which has analytical solutions, that numerical weak-form solutionscan be recovered to high order accuracy by solving a first-order reduced systemof ordinary differential equations. The method-of-lines framework is applied tothe DiscoTEX algorithm i.e through discontinuous collocation withimplicit-turned-explicit (IMTEX) integration methods which are symmetric andconserve symplectic structure. Furthermore, the main application of thealgorithm is proved, for the first-time, 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 DiscoTEX. This is shown, firstly bysolving the wave- (or wave-like) equation and comparing the numerical weak-formsolution to the exact solution. Finally, one shows how to reconstruct thescalar and gravitational metric perturbations from weak-form numericalsolutions of a non-rotating black hole, which do not have known exactanalytical solutions, and compare against state-of-the-art frequency domainresults. One concludes by motivating how DiscoTEX, and related algorithms, opena promising new alternative Extreme-Mass-Ratio-Inspiral (EMRI)s waveformgeneration route via a self-consistent evolution for the gravitationalself-force programme in the time-domain.

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