Hyperbolic Conservation Laws on Manifolds. Total Variation Estimates and the Finite Volume Method
Abstract
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 nonincreasing in time. Our results are next specialized to the important case of a flow on the 2sphere, and examples of flux are discussed. Second, we establish the convergence of the finite volume methods based on numerical fluxfunctions 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 December 2006
 arXiv:
 arXiv:math/0612847
 Bibcode:
 2006math.....12847A
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Numerical Analysis;
 35L65;
 74J40;
 58J;
 76N10
 EPrint:
 27 pages. This is Part 2 of a series