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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).


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.


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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.


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

zbMATH: 1114.35121
MathSciNet: MR2254012

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

Keywords: Entropy solution , finite volume method , hyperbolic conservation law , Riemannian manifold , Total variation

Rights: Copyright © 2005 International Press of Boston

Vol.12 • No. 3 • September 2005
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