Asymptotic Structure of Cosmological Burgers Flows in One and Two Space Dimensions: A Numerical Study (Communications in Computational Physics, 2020)
We study the cosmological Burgers model, as we call it, which is a nonlinearhyperbolic balance law (in one and two spatial variables) posed on an expanding orcontracting background. We design a finite volume scheme that is fourth-order intime and second-order in space, and allows us to compute weak solutions containingshock waves. Our main contribution is the study of the asymptotic structure of thesolutions as the time variable approaches infinity (in the expanding case) or zero (inthe contracting case). We discover that a saddle competition is taking place whichinvolves, on one hand, the geometrical effects of expanding or contracting nature and,on the other hand, the nonlinear interactions between shock waves.
Keywords: Cosmological Burgers model; shock wave; asymptotic structure; finite volume scheme; second-order accuracy; Runge-Kutta scheme.