Numerical simulations of extreme mass ratio inspirals, the most important sources for the LISA detector, face several computational challenges. We present a new approach to evolving partial differential equations occurring in black hole perturbation theory and calculations of the self-force acting on point particles orbiting supermassive black holes. Such equations are distributionally sourced, and standard numerical methods, such as finite-difference or spectral methods, face difficulties associated with approximating discontinuous functions. However, in the self-force problem we typically have access to full a priori information about the local structure of the discontinuity at the particle. Using this information, we show that high-order accuracy can be recovered by adding to the Lagrange interpolation formula a linear combination of certain jump amplitudes. We construct discontinuous spatial and temporal discretizations by operating on the corrected Lagrange formula. In a method-of-lines framework, this provides a simple and efficient method of solving time-dependent partial differential equations, without loss of accuracy near moving singularities or discontinuities. This method is well-suited for the problem of time-domain reconstruction of the metric perturbation via the Teukolsky or Regge–Wheeler–Zerilli formalisms. Parallel implementations on modern CPU and GPU architectures are discussed.

极限质量比吸气体（LISA检测器的最重要来源）的数值模拟面临若干计算挑战。我们提出了一种新的方法来演化出现在黑洞扰动理论中的偏微分方程，以及作用在绕超大质量黑洞运动的点粒子上的自力的计算。这样的方程是分布源的，并且诸如有限差分或频谱方法之类的标准数值方法面临与逼近不连续函数相关的困难。但是，在自力问题中，我们通常可以先验先验有关粒子间断的局部结构的信息。使用此信息，我们表明可以通过向Lagrange插值公式中添加某些跳跃幅度的线性组合来恢复高阶精度。我们通过对校正后的拉格朗日公式进行运算来构造不连续的空间和时间离散。在线法框架中，这提供了一种简单有效的方法，可以求解与时间有关的偏微分方程，而不会在运动奇异点或不连续点附近降低精度。此方法非常适合通过Teukolsky或Regge-Wheeler-Zerilli形式主义对度量摄动进行时域重构的问题。讨论了在现代CPU和GPU架构上的并行实现。