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.

翻译：本文提出了一种快速、高精度时域求解器，用于求解克尔时空超曲面叶状结构上的标量、电磁和引力扰动。传统时域Teukolsky方程求解器通常采用显式方法，这类方法会在数值上破坏诺特对称性，并受库朗特条件限制。当模拟预期在LISA波段持续2-5年的长时间极端质量比旋进时，这些限制会制约显式格式的性能。为此，我们探索了对称（指数型、Padé型或Hermite型）积分器，这类积分器无条件稳定，且已知能保持某些诺特对称性和相空间体积。对于线性双曲方程，这些隐式积分器可化为显式形式，使其特别适用于长时间的黑洞扰动演化。采用多项式配置方法对1+1模态Teukolsky方程进行空间离散，将其约化为通过模耦合数组和差分算子矩阵耦合的常微分方程线性系统。我们利用矩阵化技术将模耦合系统转化为适合线方法的框架形式，这一转化简化了数值实现，并能在CPU和GPU架构上实现高效并行化。通过研究亚极端和极端情形下克尔时空扰动的晚期拖尾行为，检验了我们的数值代码。