High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes

Abstract

We present a new family of very high order accurate direct Arbitrary-Lagrangian-Eulerian (ALE) Finite Volume (FV) and Discontinuous Galerkin (DG) schemes for the solution of nonlinear hyperbolic PDE systems on moving two-dimensional Voronoi meshes that are regenerated at each time step and which explicitly allow topology changes in time. The Voronoi tessellations are obtained from a set of generator points that move with the local fluid velocity. We employ an AREPO-type approach [173], which rapidly rebuilds a new high quality mesh exploiting the previous one, but rearranging the element shapes and neighbors in order to guarantee that the mesh evolution is robust even for vortex flows and for very long simulation times. The old and new Voronoi elements associated to the same generator point are connected in space–time to construct closed space–time control volumes, whose bottom and top faces may be polygons with a different number of sides. We also have to incorporate some degenerate space–time sliver elements, which are needed in order to fill the space–time holes that arise because of the topology changes in the mesh between time tn and time tn+1. The final ALE FV-DG scheme is obtained by a novel redesign of the high order accurate fully discrete direct ALE schemes of Boscheri and Dumbser [21,23], which have been extended here to general moving Voronoi meshes and space–time sliver elements. Our new numerical scheme is based on the integration over arbitrary shaped closed space–time control volumes combined with a fully-discrete space–time conservation formulation of the governing hyperbolic PDE system. In this way the discrete solution is conservative and satisfies the geometric conservation law (GCL) by construction. Numerical convergence studies as well as a large set of benchmark problems for hydrodynamics and magnetohydrodynamics (MHD) demonstrate the accuracy and robustness of the proposed method. Our numerical results clearly show that the new combination of very high order schemes with regenerated meshes that allow topology changes in each time step lead to substantial improvements compared to direct ALE methods on moving conforming meshes without topology change.