%0 Journal Article
%T Hyperbolic Conservation Laws on Manifolds. Total Variation Estimates and the Finite Volume Method
%A Paulo Amorim
%A Matania Ben-Artzi
%A Philippe G. LeFloch
%J Mathematics
%D 2006
%I arXiv
%X This paper investigates some properties of entropy solutions of hyperbolic conservation laws on a Riemannian manifold. First, we generalize the Total Variation Diminishing (TVD) property to manifolds, by deriving conditions on the flux of the conservation law and a given vector field ensuring that the total variation of the solution along the integral curves of the vector field is non-increasing in time. Our results are next specialized to the important case of a flow on the 2-sphere, and examples of flux are discussed. Second, we establish the convergence of the finite volume methods based on numerical flux-functions satisfying monotonicity properties. Our proof requires detailed estimates on the entropy dissipation, and extends to general manifolds an earlier proof by Cockburn, Coquel, and LeFloch in the Euclidian case.
%U http://arxiv.org/abs/math/0612847v1