DiscoTEX 1.0: Discontinuous collocation and implicit-turned-explicit (IMTEX) integration symplectic, symmetric numerical algorithms with high order jumps for differential equations II: extension to higher-orders of numerical convergence

TL;DR

The aim of this second paper is to present DiscoTEX's extension up to twelve orders, by computing numerical weak-form solutions to the distributionally sourced wave equation and comparing it to its exact solutions.

Abstract

\texttt{DiscoTEX} is a highly accurate numerical algorithm for computing numerical weak-form solutions to distributionally sourced partial differential equations (PDE)s. The aim of this second paper, succeeding \cite{da2024discotex}, is to present its extension up to twelve orders. This will be demonstrated by computing numerical weak-form solutions to the distributionally sourced wave equation and comparing it to its exact solutions. The full details of the numerical scheme at higher orders will be presented.

Figures (5)

• Figure 1: Numerical error associated with the numerical evaluation of the integral in equation \ref{['ch3_legendreQP_time']} with the numerical scheme of equations (\ref{['app3_disco_time_h2']} - \ref{['app3_disco_time_Jh10']}) corrected by incorporating the discontinuous behaviour through the discontinuous time-integration rules given through equations (\ref{['app3_disco_time_Jh2']} - \ref{['app3_disco_time_Jh12']}). For numerical integration schemes of 2nd to 12th order numerical convergence in the same order is observed as given by the lines DH2-DH12. Inaccurate results are recorded for a smooth integrator as given by line SH2-SH12. The computational wall-clock times for the simulations are given in Table \ref{['tab_discontinuousTime-allorders']}.
• Figure 2: Numerical weak-form solution to $\Psi(\tau,\sigma)$ obtained via the DiscoTEX H66th- order algorithm. Left: Numerical field $\Psi(\tau,\sigma)$ for a point-particle in time-dependent linear motion $\xi_{p}(\tau_{c})$, where $v$ is the particle's velocity. Specifically, here $\xi_{p} \approx 0.608$ at the coordinate-time $\tau_{c} \approx0.876$ and $v=1/4$. Right: Waveform for the point-particle computed on the numerical domain $\sigma \in [0,1]$ and $\tau \in [-1.52, 4.50]$. As expected the result matches that observed for the solution obtained with DiscoTEX H44th-order numerical algorithm da2024discotex.
• Figure 3: Phase portrait for the numerical weak-form solution obtained via DiscoTEX H6 with a 6th- order IMTEX Hermite H6 time integrator in the $(\tau,\sigma)$ hyperboloidal chart. The numerical weak-form solutions to the field $\Psi(\tau,\sigma_{f})$, $\Pi(\tau,\sigma_{f})$ are evaluated in the numerical time interval $\tau \in [-1.52,4.50]$ at the last grid point. As reminded in da2024discotex here we too point out that unlike in da2023hyperboloidal, Figure 2, uses the exact solutions as initial data, whereas in our previous work, we used trivial initial data, thus not requiring the monitoring of an extra user-specifiable control factor.
• Figure 4: Numerical convergence studies assessing the optimal user-specifiable control factors: number of N nodes - [CTRL F\ref{['controlfactor1']}] (top plot) and number of J jumps - [CTRL F\ref{['controlfactor2']}] (bottom plot). Whereas there is a significant accuracy improvement after the second-order time-integration scheme, the difference between the 4th and 6th time-stepper is marginal for a similar numerical set-up. It is reasonable to pick N=45 Chebyshev collocation nodes and J=19 jumps.
• Figure 5: Numerical error associated with computation of the numerical weak-form solution $\Psi(\tau,\sigma_{f})$ against the exact solution in equation \ref{['weakformsolution_wave']} with both the discontinuous Hermite integrator of order-2, order-4 and order-6, respectively DH2,DH4, DH6. As expected, we observe in lines DH2, DH4, DH6 that the DiscoTEX algorithm converges to its expected orders respectively.