We present a fourth-order convergent, (2+1)-dimensional, numerical formalism to solve the Teukolsky equation in the time domain. Our approach is first to rewrite the Teukolsky equation as a system of first-order differential equations. In this way we get a system that has the form of an advection equation. This is then used in combination with a series expansion of the solution in powers of time. To obtain a fourth-order scheme we kept terms up to fourth derivative in time and use the advectionlike system of differential equations to substitute the temporal derivatives by spatial derivatives. This scheme is applied to evolve gravitational perturbations in the Schwarzschild and Kerr backgrounds. Our numerical method proved to be stable and fourth-order convergent in r∗ and θ directions. The correct power-law tail, ∼1/t2ℓ+3, for general initial data, and ∼1/t2ℓ+4, for time-symmetric data, was found in our runs. We noted that it is crucial to resolve accurately the angular dependence of the mode at late times in order to obtain these values of the exponents in the power-law decay. In other cases, when the decay was too fast and round-off error was reached before a tail was developed, then the quasinormal modes frequencies provided a test to determine the validity of our code.
Pazos-Avalos, Enrique, and Carlos O. Lousto. "Numerical integration of the Teukolsky equation in the time domain." Physical Review D 72.8 (2005): 084022. https://doi.org/10.1103/PhysRevD.72.084022
Physical Review D