Geometric Error of Finite Volume Schemes for Conservation Laws on Evolving Surfaces

TL;DR

This paper compares theoretical and practical finite volume schemes for conservation laws on evolving surfaces, proving their solutions converge at the same rate and confirming optimal convergence through numerical experiments.

Contribution

It establishes that practical schemes on moving polyhedra approximate the theoretical schemes with the same convergence rate, enabling feasible computations on complex geometries.

Findings

Practical schemes converge to the entropy solution at the same rate as theoretical schemes.

Error between schemes is proportional to the mesh width.

Numerical experiments confirm the optimal order of convergence.

Abstract

This paper studies finite volume schemes for scalar hyperbolic conservation laws on evolving hypersurfaces of $\mathbb{R}^3$. 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.