Moving-Mesh Finite-Volume Methods for Hyperbolic Interface Dynamics

The numerical discretization of continuum-mechanical free boundary value problems for hyperbolic conservation laws becomes challenging when the dynamics of the interface depend sensitively on smaller-scale effects. A proper tracking of the interface and an efficient solution of the conservation laws in the bulk domains can be realized by a heterogeneous multi-scale ansatz combined with recently introduced moving-mesh concepts for finite-volume methods. To illustrate the approach we focus on two applications: the tracking of phase boundaries in compressible liquid-vapour flow and dimensionally mixed models for two-phase flow in fractured porous media. In the first case phase transition effects lead to non-standard interface dynamics. In the latter case the coupling conditions for the bulk domains involve the solution of evolution equations in the fractures which are represented as hypersurfaces.