2012/06/14 by Philippe G. LeFloch, LeFloch, Philippe G., Hasan Makhlof +3 · 1 citation
Engineering · Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #Computational Fluid Dynamics and Aerodynamics #Cosmology and Gravitation Theories #FOS: Mathematics #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1206.3018
openalex publication_date 2012/06/14 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
Within the class of nonlinear hyperbolic balance laws posed on a curved\nspacetime (endowed with a volume form), we identify a hyperbolic balance law\nthat enjoys the same Lorentz invariance property as the one satisfied by the\nEuler equations of relativistic compressible fluids. This model is unique up to\nnormalization and converges to the standard inviscid Burgers equation in the\nlimit of infinite light speed. Furthermore, from the Euler system of\nrelativistic compressible flows on a curved background, we derive, both, the\nstandard inviscid Burgers equation and our relativistic generalizations. The\nproposed models are referred to as relativistic Burgers equations on curved\nspacetimes and provide us with simple models on which numerical methods can be\ndeveloped and analyzed. Next, we introduce a finite volume scheme for the\napproximation of discontinuous solutions to these relativistic Burgers\nequations. Our scheme is formulated geometrically and is consistent with the\nnatural divergence form of the balance laws under consideration. It applies to\nweak solutions containing shock waves and, most importantly, is well-balanced\nin the sense that it preserves steady solutions. Numerical experiments are\npresented which demonstrate the convergence of the proposed finite volume\nscheme and its relevance for computing entropy solutions on a curved\nbackground.\n