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


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Lefloch P. G. , Okutmustur B. , Neves W.

ACTA MATHEMATICA SINICA-ENGLISH SERIES, vol.25, pp.1041-1066, 2009 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 25
  • Publication Date: 2009
  • Doi Number: 10.1007/s10114-009-8090-y
  • Title of Journal : ACTA MATHEMATICA SINICA-ENGLISH SERIES
  • Page Numbers: pp.1041-1066
  • Keywords: Hyperbolic conservation law, entropy solution, finite volume scheme, error estimate, discrete entropy inequality, convergence rate, DIFFERENCE SCHEMES, SPACE DIMENSIONS, CONVERGENCE

Abstract

Following Ben-Artzi and LeFloch, we consider nonlinear hyperbolic conservation laws posed on a Riemannian manifold, and we establish an L1-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 L1 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 extent the arguments to curved manifolds, by taking into account the effects to the geometry and overcoming several new technical difficulties.

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.