Relativistic Burgers Equations on Curved Spacetimes. Derivation and Finite Volume Approximation

Abstract

Within the class of nonlinear hyperbolic balance laws posed on a curved spacetime (endowed with a volume form), we identify a hyperbolic balance law that enjoys the same Lorentz invariance property as the one satisfied by the Euler equations of relativistic compressible fluids. This model is unique up to normalization and converges to the standard inviscid Burgers equation in the limit of infinite light speed. Furthermore, from the Euler system of relativistic compressible flows on a curved background, we derive both the standard inviscid Burgers equation and our relativistic generalizations. The proposed models are referred to as relativistic Burgers equations on curved spacetimes and provide us with simple models on which numerical methods can be developed and analyzed. Next, we introduce a finite volume scheme for the approximation of discontinuous solutions to these relativistic Burgers equations. Our scheme is formulated geometrically and is consistent with the natural divergence form of the balance laws under consideration. It applies to weak solutions containing shock waves and, most importantly, is well balanced in the sense that it preserves steady solutions. Numerical experiments are presented which demonstrate the convergence of the proposed finite volume scheme and its relevance for computing entropy solutions on a curved background.

Finite volume approximation of the relativistic Burgers equation on a Schwarzschild--(anti-)de Sitter spacetime

“…We notice that, from the pressureless Euler system on flat spacetime, one can recover the nonrelativistic (classical) Burgers equation (see [6]). …”

“…We are inspired by the work of LeFloch and his collaborators for the derivation of the model and we apply the strategy therein [2,3,6]. Similar to the nonrelativistic case, we impose zero pressure in the Euler equations (1.1) on the SdS-SAdS spacetimes to derive the relativistic Burgers equation.…”

“…• If Λ = 0, (2.12) reduces (after a suitable normalization) to the vanishing pressure Euler system on the Schwarzschild background (see [6]). …”

“…The theory of derivation of relativistic-type Burgers equations on curved spacetimes was initiated by LeFloch et al [6] and developed together with collaborators [2][3][4]. In particular, the first model of interest, which may be considered as the fundamental relativistic Burgers equation, was derived in different ways, either from the Euler system on a curved background or by introducing a hyperbolic balance law on a flat background carrying out the same Lorentz invariance property fulfilled by the Euler equations of relativistic compressible fluids.…”

“…In particular, the first model of interest, which may be considered as the fundamental relativistic Burgers equation, was derived in different ways, either from the Euler system on a curved background or by introducing a hyperbolic balance law on a flat background carrying out the same Lorentz invariance property fulfilled by the Euler equations of relativistic compressible fluids. This analysis has recently been extended to the Schwarzschild, Friedmann-Lemaitre-Robertson-Walker (FLRW), de Sitter, and anti-de Sitter spacetimes and studied numerically by means of finite volume approximation [2][3][4]6].…”