Asymptotic structure of cosmological Burgers flows in one and two space dimensions: a numerical study

TL;DR

This paper investigates the long-term behavior of solutions to the cosmological Burgers model in one and two dimensions, using a high-order numerical scheme to analyze shock wave interactions on expanding or contracting backgrounds.

Contribution

It introduces a fourth-order finite volume scheme for the cosmological Burgers model and studies the asymptotic structure of solutions in expanding and contracting scenarios.

Findings

Identification of a saddle competition between geometric effects and shock interactions.

Analysis of asymptotic solution behavior as time approaches infinity or zero.

Numerical evidence of the interplay between expansion/contraction and nonlinear shocks.

Abstract

We study the cosmological Burgers model, as we call it, which is a nonlinear hyperbolic balance law (in one and two spatial variables) posed on an expanding or contracting background. We design a finite volume scheme that is fourth-order in time and second-order in space, and allows us to compute weak solutions containing shock waves. Our main contribution is the study of the asymptotic structure of the solutions as the time variable approaches infinity (in the expanding case) or zero (in the contracting case). We discover that a saddle competition is taking place which involves, on one hand, the geometrical effects of expanding or contracting nature and, on the other hand, the nonlinear interactions between shock waves.