A convergence result for finite volume schemes on riemannian manifolds

Giesselmann, Jan

doi:10.1051/m2an/2009013

Geodesic

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A convergence result for finite volume schemes on riemannian manifolds

Giesselmann, Jan

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 43 (2009) no. 5, pp. 929-955.

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Résumé

This paper studies a family of finite volume schemes for the hyperbolic scalar conservation law $u_{t} + \nabla_{g} \cdot f (x, u) = 0$ on a closed riemannian manifold $M .$ For an initial value in BV( $M$ ) we will show that these schemes converge with a $h^{\frac{1}{4}}$ convergence rate towards the entropy solution. When $M$ is $1$ -dimensional the schemes are TVD and we will show that this improves the convergence rate to $h^{\frac{1}{2}} .$

MR Zbl

DOI : 10.1051/m2an/2009013

Classification : 74S10, 35L65, 58J45
Mots-clés : finite volume method, conservation law, curved manifold

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     author = {Giesselmann, Jan},
     title = {A convergence result for finite volume schemes on riemannian manifolds},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {929--955},
     publisher = {EDP-Sciences},
     volume = {43},
     number = {5},
     year = {2009},
     doi = {10.1051/m2an/2009013},
     zbl = {1173.74454},
     mrnumber = {2559739},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/articles/10.1051/m2an/2009013/}
}

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Giesselmann, Jan. A convergence result for finite volume schemes on riemannian manifolds. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 43 (2009) no. 5, pp. 929-955. doi : 10.1051/m2an/2009013. https://geodesic-test.mathdoc.fr/articles/10.1051/m2an/2009013/

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