A spectral collocation approximation for the radial-infall of a compact object into a Schwarzschild black hole

Abstract

The inhomogeneous Zerilli equation is solved in time-domain numerically with the Chebyshev spectral collocation method to investigate a radial-infall of the point particle towards a Schwarzschild black hole. Singular source terms due to the point particle appear in the equation in the form of the Dirac $\delta$-function and its derivative. For the approximation of singular source terms, we use the direct derivative projection method without any regularization. The gravitational waveforms are evaluated as a function of time. We compare the results of the spectral collocation method with those of the explicit second-order central-difference method. The numerical results show that the spectral collocation approximation with the direct projection method is accurate and converges rapidly when compared with the finite-difference method.

Accepted for publication in International Journal of Modern Physics C (IJMPC)

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• Fourth order indirect integration method for black hole perturbations: even modes
• Discontinuous Galerkin method for linear wave equations involving derivatives of the Dirac delta distribution