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

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 extend the arguments to curved manifolds, by taking into account the effects to the geometry and overcoming several new technical difficulties.