Abstract. We investigate the numerical approximation of (discontinuous) entropy solutions to nonlinear hyperbolic conservation laws posed on a Lorentzian manifold. Our main result establishes the convergence of monotone and first-order finite volume schemes for a large class of (space and time) triangulations. The proof relies on a discrete version of entropy inequalities and an entropy dissipation bound, which take into account the manifold geometry and were originally discovered by Cockburn, Coquel, and LeFloch in the (flat) Euclidian setting.Key words. Conservation law, Lorenzian manifold, entropy condition, measure-valued solution, finite volume scheme, convergence analysis.AMS subject classifications. Primary: 35L65; Secondary: 76L05, 76N. IntroductionWe are interested in discontinuous solutions to nonlinear hyperbolic conservation laws posed on a globally hyperbolic Lorentzian manifold, and we introduce a class of first-order, monotone finite volume schemes which enjoy geometrically natural stability properties. In turn, we conclude that the proposed finite volume schemes converge (in a strong topology) toward entropy solutions to hyperbolic conservation laws. Recall that the well-posedness theory for nonlinear hyperbolic equations posed on a manifold was recently established by Ben-Artzi and LeFloch [2] and LeFloch and Okutmustur [17,18]. On the other hand, our proof of convergence of the finite volume method can be viewed as a generalization to Lorentzian manifolds of the technique introduced by Cockburn, Coquel and LeFloch [5,7] for the (flat) Euclidean setting and already extended to Riemannian manifolds by Amorim, Ben-Artzi, and LeFloch [1].Major conceptual and technical difficulties arise in the analysis of partial differential equations posed on a Lorentzian manifold. Several new difficulties also appear when trying to generalize the convergence results in [1,5,7] to Lorentzian manifolds. Most importantly, a space and time triangulation must be introduced and the geometry of the manifold must be taken into account in the discretization. We point out that, on a Lorentzian manifold, one cannot canonically choose a preferred foliation by spacelike hypersurfaces in general, so that it is important for the discretization to be robust enough to allow for a large class of foliations and of spacetime triangulations. From the numerical analysis standpoint, it is challenging to design and analyze discretization schemes that are consistent with the geometry of the given manifold. Our guide in deriving the necessary estimates was to ensure that all of our arguments are intrinsic in nature, and thus do not explicitly rely on a choice of local coordinates.The main assumption of global hyperbolicity made on the given Lorentzian background is natural, and ensures that the manifold enjoys reasonable causality properties. Furthermore, the class of schemes considered in the present paper is quite general
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.
Following Ben-Artzi and LeFloch, we consider nonlinear hyperbolic conservation laws posed on a Riemannian manifold, and we establish an L 1 -error estimate for a class of finite volume schemes allowing for the approximation of entropy solutions to the initial value problem. The error in the L 1 norm is of order h 1/4 at most, where h represents the maximal diameter of elements in the family of geodesic triangulations. The proof relies on a suitable generalization of Cockburn, Coquel, and LeFloch's theory which was originally developed in the Euclidian setting. We extent the arguments to curved manifolds, by taking into account the effects to the geometry and overcoming several new technical difficulties.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.