A moving mesh finite volume method for hyperbolic interface problems

Abstract

Physical processes involving multiple phases naturally give rise to interface problems. We present a novel method in multiple space dimensions to approximate weak solutions of systems of hyperbolic or hyperbolic-elliptic conservation laws. The method is a combination of the Finite Volume method with moving meshes, which is capable of tracking dynamic interfaces, including undercompressive shock waves. We prove that the method is conservative and recovers planar discontinuous waves exactly. For the sharp resolution of the interface, an interface-preserving moving mesh algorithm in two or higher dimensions is presented. It resolves a moving (𝑑-1)-dimensional manifold directly within the 𝑑-dimensional mesh, which means that the interface is represented by a subset of moving mesh cell surfaces. The local remeshing algorithms allow for strongminterface deformations. We prove that the given algorithms preserve the interface after interface deformation and remeshing steps. To demonstrate the efficiency and reliability of the new method, we test it in two and three space dimensions for scalar model problems, for compressible liquid-vapor flow, and for the anti-plane shear deformation model. Furthermore, we derive an exact Riemann solver for the two-dimensional anti-plane shear deformation model as a special case of the elasticity equations. This new Riemann solver is used and validated in numerical experiments.