WIAS Preprint No. 1760, (2013)

Geometric error of finite volume schemes for conservation laws on evolving surfaces



Authors

  • Giesselmann, Jan
  • Müller, Thomas

2010 Mathematics Subject Classification

  • 65M08 35L65 58J45

Keywords

  • hyperbolic conservation laws, finite volume schemes, curved surfaces, error bound

Abstract

This paper studies finite volume schemes for scalar hyperbolic conservation laws on evolving hypersurfaces of $mathbbR^3$. We compare theoretical schemes assuming knowledge of all geometric quantities to (practical) schemes defined on moving polyhedra approximating the surface. For the former schemes error estimates have already been proven, but the implementation of such schemes is not feasible for complex geometries. The latter schemes, in contrast, only require (easily) computable geometric quantities and are thus more useful for actual computations. We prove that the difference between approximate solutions defined by the respective families of schemes is of the order of the mesh width. In particular, the practical scheme converges to the entropy solution with the same rate as the theoretical one. Numerical experiments show that the proven order of convergence is optimal.

Appeared in

  • Numerische Mathematik 128 (2014), pp. 489--516.

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