  We consider an isothermal compressible fluid evolving on a cosmological
background which may be 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 in two
space dimensions are performed in order to support our observations.
