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This work outlines a fast, high-precision time-domain solver for scalar,electromagnetic and gravitational perturbations on hyperboloidal foliations ofKerr space-times. Time-domain Teukolsky equation solvers have typically usedexplicit methods, which numerically violate Noether symmetries and areCourant-limited. These restrictions can limit the performance of explicitschemes when simulating long-time extreme mass ratio inspirals, expected toappear in LISA band for 2-5 years. We thus explore symmetric (exponential,Pad\'e or Hermite) integrators, which are unconditionally stable and known topreserve certain Noether symmetries and phase-space volume. For linearhyperbolic 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 polynomialcollocation methods and reduced to a linear system of ordinary differentialequations, coupled via mode-coupling arrays and discretized (matrix)differential operators. We use a matricization technique to cast themode-coupled system in a form amenable to a method-of-lines framework, whichsimplifies numerical implementation and enables efficient parallelization onCPU and GPU architectures. We test our numerical code by studying late-timetails of Kerr spacetime perturbations in the sub-extremal and extremal cases.

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