Second-order hyperbolic Fuchsian systems. Gowdy spacetimes and the Fuchsian numerical algorithm

TL;DR

This paper develops a numerical algorithm based on Fuchsian methods for solving the singular initial value problem in second-order hyperbolic Fuchsian systems, specifically applied to Gowdy spacetimes in general relativity.

Contribution

It introduces a Fuchsian numerical algorithm for accurate approximation of solutions to singular hyperbolic systems, applied to Einstein's equations for Gowdy spacetimes.

Findings

The algorithm achieves high accuracy in numerical simulations.

It efficiently constructs Gowdy spacetimes with Cauchy horizons.

The method simplifies previous well-posedness proofs.

Abstract

This is the second part of a series devoted to the singular initial value problem for second-order hyperbolic Fuchsian systems. In the first part, we defined and investigated this general class of systems, and we established a well-posedness theory in weighted Sobolev spaces. This theory is applied here to the vacuum Einstein equations for Gowdy spacetimes admitting, by definition, two Killing fields satisfying certain geometric conditions. We recover, by more direct and simpler arguments, the well-posedness results established earlier by Rendall and collaborators. In addition, in this paper we introduce a natural approximation scheme, which we refer to as the Fuchsian numerical algorithm and is directly motivated by our general theory. This algorithm provides highly accurate, numerical approximations of the solution to the singular initial value problem. In particular, for the class of Gowdy spacetimes under consideration, various numerical experiments are presented which show the interest and efficiency of the proposed method. Finally, as an application, we numerically construct Gowdy spacetimes containing a smooth, incomplete, non-compact Cauchy horizon.