A convergence result for finite volume schemes on riemannian manifolds

Giesselmann, Jan

ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 5, pp. 929-955

This paper studies a family of finite volume schemes for the hyperbolic scalar conservation law $u_t+{\nabla }_g·f(x,u)=0$ 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 $h^{\frac{1}{2}}.$

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

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