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Asymptotic structure of cosmological fluid flows: a numerical study

Yangyang Cao, Mohammad A. Ghazizadeh and Philippe G. LeFloch

Vol. 17 (2022), No. 1, 79–129

We consider an isothermal compressible fluid evolving on a cosmological background which is either expanding or contracting toward the future. The Euler equations governing such a flow consist of two nonlinear hyperbolic balance laws which we treat in one and in two space dimensions. We design a finite volume scheme which is fourth-order accurate in time and second-order accurate in space. This scheme allows us to compute weak solutions containing shock waves and, by design, is well-balanced in the sense that it preserves exactly a special class of solutions. Using this scheme, we investigate the asymptotic structure of the fluid when the time variable approaches infinity (in the expanding regime) or approaches zero (in the contracting regime). We study these two limits by introducing a suitable rescaling of the density and velocity variables and, in turn, we analyze the effects induced by the geometric terms (of expanding or contracting nature) and the nonlinear interactions between shocks. Extensive numerical experiments in one and two space dimensions are performed in order to support our observations.

cosmological Euler model, shock wave, asymptotic structure, finite volume scheme, geometry-preserving, high-order accuracy
Mathematical Subject Classification 2010
Primary: 76L05, 76M12
Secondary: 83F05
Received: 16 February 2020
Revised: 24 November 2021
Accepted: 12 December 2021
Published: 7 October 2022
Yangyang Cao
Laboratoire Jacques-Louis Lions
Sorbonne Université
75252 Paris
Department of Mathematics
Southern University of Science and Technology
518055 Shenzhen
Mohammad A. Ghazizadeh
Department of Civil Engineering
University of Ottawa
Ottawa ON K1N 6N5
Philippe G. LeFloch
Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique
Sorbonne Université
75252 Paris