The relativistic Burgers equation on a de Sitter spacetime. Derivation and finite volume approximation

Abstract

The inviscid Burgers equation is one of the simplest nonlinear hyperbolic conservation law which provides a variety examples for many topics in nonlinear partial differential equations such as wave propagation, shocks and perturbation, and it can easily be derived by the Euler equations of compressible fluids by imposing zero pressure in the given system. Recently, several versions of the relativistic Burgers equations have been derived on different geometries such as Minkowski (flat), Schwarzshild and FLRW spacetimes by LeFloch and his collaborators. In this paper, we consider a family member of the FLRW spacetime so-called the de Sitter background, introduce some important features of this spacetime geometry with its metric and derive the relativistic Burgers equation on it. The Euler system of equations on the de Sitter spacetime can be found by a known process by using the Christoffel symbols and tensors for perfect fluids. We applied the usual techniques used for the Schwarzshild and FLRW spacetimes in order to derive the relativistic Burgers equation from the vanishing pressure Euler system on the de Sitter background. We observed that the model admits static solutions. In the final part, we examined several numerical illustrations of the given model through a finite volume approximation based on the paper by LeFloch et al. The effect of the cosmological constant is also numerically analysed in this part. Furthermore, a comparison of the static solution with the Lax Friedrichs scheme is implemented so that the results demonstrate the efficiency and robustness of the finite volume scheme for the derived model