Hyperbolic conservation laws on manifolds. An error estimate for finite volume schemes

Abstract

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.

Hyperbolic conservation laws on the sphere. A geometry-compatible finite volume scheme

“…The theoretical background about the well-posedness theory for hyperbolic conservation laws on manifolds was established recently by Ben-Artzi and LeFloch [6] together with collaborators [1,2,8]. An important condition arising in the theory is the "zerodivergence" or geometric-compatibility property of the flux vector; a basic requirement in our construction of a finite volume scheme is to formulate and ensure a suitable discrete version of this condition.…”

Traces for Functions of Bounded Variation on Manifolds with Applications to Conservation Laws on Manifolds with Boundary

“…To our knowledge, conservation laws on manifolds were studied for the first time by [24] showing existence and uniqueness for a geometryindependent variant of (1.1). For the case of compact manifolds without boundary the theoretical and numerical study of conservation laws on manifolds has reached significant progress [8,2,7,3,19,11,20,13,14] during the last decade. The Dirichlet problem for a geometry-independent formulation of (1.1) was addressed by Panov using a kinetic formulation.…”

Well-posedness theory for stochastically forced conservation laws on Riemannian manifolds

“…shallow water waves on the surface of a planet (caricature model of the atmosphere), and general relativity in which the Einstein-Euler equations are posed on a manifold with the metric being one of the unknowns. For scalar conservation laws defined on manifolds, the development of a theory of well-posedness and numerical approximations (of Kružkov-DiPerna solutions) was initiated by LeFloch and co-authors [1,2,6,7,8,44,45,46] (see also Panov [52,53]). The subject has been extended in several directions by different authors, including Giesselmann [30], Dziuk, Kröner, and Müller [24], Lengeler and Müller [47], Giesselmann and Müller [31], and Kröner, Müller, and Strehlau [40], and Graf, Kunzinger, and Mitrovic [32].…”