A numerical algorithm for Fuchsian equations and fluid flows on cosmological spacetimes

TL;DR

This paper develops a numerical method for solving Fuchsian equations related to fluid flows on cosmological spacetimes, analyzing errors and demonstrating stability in a specific cosmological model.

Contribution

It introduces a novel numerical algorithm for Fuchsian equations with singular initial data, combining spectral and Runge-Kutta methods, and provides a detailed error analysis.

Findings

Numerical experiments confirm the theoretical error estimates.

The method effectively models fluid flows on Kasner spacetime.

Stability of flows in the sub-critical regime is demonstrated.

Abstract

We consider a class of Fuchsian equations that, for instance, describes the evolution of compressible fluid flows on a cosmological spacetime. Using the method of lines, we introduce a numerical algorithm for the singular initial value problem when data are imposed on the cosmological singularity and the evolution is performed from the singularity hypersurface. We approximate the singular Cauchy problem of Fuchsian type by a sequence of regular Cauchy problems, which we next discretize by pseudo-spectral and Runge-Kutta techniques. Our main contribution is a detailed analysis of the numerical error which has two distinct sources, and our main proposal here is to keep in balance the errors arising at the continuum and at the discrete levels of approximation. We present numerical experiments which strongly support our theoretical conclusions. This strategy is finally applied to applied to compressible fluid flows evolving on a Kasner spacetime, and we numerically demonstrate the nonlinear stability of such flows, at least in the so-called sub-critical regime identified earlier by the authors.