Stability in Wave Equation on Schwarzschild de-Sitter Spacetime

Anyone Walking into my room can see me reading a lot of Research Papers on Arxiv. So, these papers[Find them in the Reference Section of the Paper] I was reading these days, so thought to prepare some notes and add some views

Abstract

We explore numerical stability issues and hyperboloidal coordinate choices for solving the wave equation on Schwarzschild-de Sitter (SdS) spacetime. In particular, we consider the treatment of the origin with a spherical symmetry boundary condition and use finite differences to discretize the equation. We demonstrate that a direct discretization is numerically unstable and propose an alternative stable approach by employing a suitable identity. We then introduce hyperboloidal coordinates and explore different choices for the height function to remove the metric singularity at the roots of $f$, which is the metric function in the SdS spacetime. We analyze the characteristic speeds of spherical light rays and assess the numerical properties of each coordinate choice. Finally, we present a regularized form of the scalar wave equation after transformations into tortoise and compactified hyperboloidal coordinates. This study provides valuable insights into numerical stability and coordinate choices for solving wave equations on SdS spacetime, paving the way for reliable simulations of gravitational wave phenomena and other physics in this spacetime background.