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

TL;DR

This paper develops a spectral collocation method using Chebyshev polynomials to accurately simulate the radial infall of a point particle into a Schwarzschild black hole, efficiently handling singular source terms.

Contribution

It introduces a direct derivative projection approach for singular sources in spectral methods, demonstrating improved accuracy and convergence over finite-difference techniques.

Findings

Spectral collocation method achieves high accuracy in modeling gravitational waveforms.

The approach converges rapidly compared to traditional finite-difference methods.

The method effectively handles Dirac delta singularities without regularization.

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.