标题： Kerr时空的双曲叶层上1+1 Teukolsky方程的对称积分 显示英文标题

标题： Symmetric integration of the 1+1 Teukolsky equation on hyperboloidal foliations of Kerr spacetimes

摘要： 本工作概述了一种快速、高精度的时间域求解器，用于Kerr时空的双曲叶层上的标量、电磁和引力扰动。 时间域Teukolsky方程求解器通常使用显式方法，这些方法在数值上会破坏Noether对称性，并受到Courant限制。 这些限制可能在模拟长时间极端质量比吸积时限制显式方案的性能，这种现象预计会在LISA频段出现2至5年。 因此，我们探索了对称（指数、Padé或Hermite）积分器，这些积分器是无条件稳定的，并且已知可以保持某些Noether对称性和相空间体积。 对于线性双曲方程，这些隐式积分器可以转换为显式形式，使其非常适合黑洞扰动的长时间演化。 1+1模态Teukolsky方程在空间上使用多项式配点方法进行离散化，并通过模式耦合数组和离散化的（矩阵）微分算子转化为一组耦合的常微分方程。 我们使用矩阵化技术将模式耦合系统转换为适用于线法框架的形式，这简化了数值实现，并在CPU和GPU架构上实现了高效的并行化。 我们通过研究亚极端和极端情况下的Kerr时空扰动的晚期尾部来测试我们的数值代码。 显示英文摘要

摘要： This work outlines a fast, high-precision time-domain solver for scalar, electromagnetic and gravitational perturbations on hyperboloidal foliations of Kerr space-times. Time-domain Teukolsky equation solvers have typically used explicit methods, which numerically violate Noether symmetries and are Courant-limited. These restrictions can limit the performance of explicit schemes when simulating long-time extreme mass ratio inspirals, expected to appear in LISA band for 2-5 years. We thus explore symmetric (exponential, Pad\'e or Hermite) integrators, which are unconditionally stable and known to preserve certain Noether symmetries and phase-space volume. For linear hyperbolic equations, these implicit integrators can be cast in explicit form, making them well-suited for long-time evolution of black hole perturbations. The 1+1 modal Teukolsky equation is discretized in space using polynomial collocation methods and reduced to a linear system of ordinary differential equations, coupled via mode-coupling arrays and discretized (matrix) differential operators. We use a matricization technique to cast the mode-coupled system in a form amenable to a method-of-lines framework, which simplifies numerical implementation and enables efficient parallelization on CPU and GPU architectures. We test our numerical code by studying late-time tails of Kerr spacetime perturbations in the sub-extremal and extremal cases.