DiscoTEX 1.0：用于微分方程的高阶跳跃不连续配点与隐式转显式（IMTEX）积分辛、对称数值算法 I：数值黑洞微扰理论应用
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
AI总结：
本文提出 DiscoTEX 1.0 数值算法，通过不连续配点与隐式转显式辛对称积分求解 Dirac δ-分布源微分方程，并展示其在黑洞微扰和引力波波形建模中的应用。
摘要：
Dirac δ-分布源微分方程出现在许多动力物理系统中，从机器学习、金融、神经科学、地震学到黑洞微扰理论。这些系统缺乏精确解析解，因此最好用数值方法处理。我们描述了一种通用数值算法，该算法分别通过在不连续 Lagrange 和 Hermite 插值公式上进行操作来构造不连续的空间和时间离散化。通过求解具有解析解的分布源波动方程，我们证明通过求解一阶约化常微分方程组，可以以高阶精度恢复数值弱形式解。该方法将线方法框架应用于 DiscoTEX 算法，即通过不连续配点与隐式转显式积分方法，这些方法是对称的并保持辛结构。此外，该算法的主要应用通过计算数值网格内任意期望位置的振幅得到证明，包括在波动（或类波动）方程通过 DiscoTEX 插值而不连续的位置（及其右极限和左极限）。首先通过求解波动（或类波动）方程并将数值弱形式解与精确解进行比较来证明这一点。我们进一步演示了如何从非旋转黑洞的弱形式数值解重建引力度规微扰，这些微扰没有已知的精确解析解，并将其与最先进的频域结果进行比较。最后，我们论证了 DiscoTEX 及相关数值算法如何为模拟高度非对称双星开辟一条有前景的新波形生成途径，并补充当前的频域方法。
英文摘要：
Dirac $δ-$ 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.
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