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September 2005 Hyperbolic Conservation Laws on Manifolds: Total Variation Estimates and the Finite Volume Method
Paulo Amorim, Matania Ben-Artzi, Philippe G. LeFloch
Methods Appl. Anal. 12(3): 291-324 (September 2005).

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 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.

Citation

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Paulo Amorim. Matania Ben-Artzi. Philippe G. LeFloch. "Hyperbolic Conservation Laws on Manifolds: Total Variation Estimates and the Finite Volume Method." Methods Appl. Anal. 12 (3) 291 - 324, September 2005.

Information

Published: September 2005
First available in Project Euclid: 5 April 2007

zbMATH: 1114.35121
MathSciNet: MR2254012

Subjects:
Primary: 35L65, 58J, 74J40, 76N10

Rights: Copyright © 2005 International Press of Boston

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Vol.12 • No. 3 • September 2005
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