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

JOURNAL OF COMPUTATIONAL PHYSICS

卷:431

Article

Beyer, Florian¹  LeFloch, Philippe G.^2,3 

[1] Univ Otago, Dept Math & Stat, POB 56, Dunedin 9054, New Zealand

[2] Sorbonne Univ, Lab Jacques Louis Lions, 4 Pl Jussieu, F-75252 Paris, France

[3] Sorbonne Univ, Ctr Natl Rech Sci, 4 Pl Jussieu, F-75252 Paris, France

关键词: Fuchsian equations;    General relativity;    Asymptotic behavior;    Fluids on Kasner;    Numerical approximation;   

DOI  :  10.1016/j.jcp.2021.110145

来源: Elsevier

【 摘 要 】

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 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. (C) 2021 Elsevier Inc. All rights reserved.

【 授权许可】

Free