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

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.

Relativistic perfect fluids near Kasner singularities

“…To support our conjecture, we discuss some observations that take us beyond the rigorous result in Theorem 1.1. Other evidence supporting our conjecture has been presented previously in [14][15][16]. To begin the discussion, we assume, without loss of generality as we explain in Section 3, that max{p 1 , p 2 , p 3 } = p 3 .…”

“…This idea was put forward in [12] and then used in [3,4] to established the first existence proof of solutions of the singular initial value problem for quasilinear symmetric hyperbolic Fuchsian equations. Besides being a useful analytic technique, this approximation technique can be employed to construct numerical solutions of the singular initial value problem [12,13,15]. 2.4.…”

“…First, Remark 3.2 also applies to Theorem 3.6, and consequently this stability result automatically extends to the more general case of fluids on Kasner-scalar field spacetimes. Second, the stability of Euler fluids near Kasner singularities in the sense of Theorem 3.6 was outstanding even in the Gowdy symmetric setting; see [16] and the related numerical investigations from [15] for details.…”