Relativistic Burgers equations on curved spacetimes. Derivation and finite volume approximation

TL;DR

This paper introduces relativistic Burgers equations on curved spacetimes, deriving models that respect Lorentz invariance, and develops a finite volume scheme for their numerical approximation, including shock capturing and steady state preservation.

Contribution

The paper derives a Lorentz-invariant relativistic Burgers model on curved spacetimes and proposes a geometric finite volume scheme for its numerical solution.

Findings

The finite volume scheme converges for solutions with shocks.

The scheme preserves steady solutions (well-balanced).

Numerical experiments validate the scheme's effectiveness on curved backgrounds.

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.