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

• Published in: Numerische Mathematik Vol. 128; no. 3; pp. 489 - 516
• Main Authors: Giesselmann, Jan, Müller, Thomas
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2014
• Subjects: Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Numerical Analysis; Numerical and Computational Physics; Simulation; Theoretical; 65M08; 58J45; 35L65
• ISSN: 0029-599X; 0945-3245
• DOI: 10.1007/s00211-014-0621-5

This paper studies finite volume schemes for scalar hyperbolic conservation laws on evolving hypersurfaces of R 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.