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We present a new family of very high order accurate directArbitrary-Lagrangian-Eulerian (ALE) Finite Volume (FV) and DiscontinuousGalerkin (DG) schemes for the solution of nonlinear hyperbolic PDE systems onmoving 2D Voronoi meshes that are regenerated at each time step and whichexplicitly allow topology changes in time.  The Voronoi tessellations are obtained from a set of generator points thatmove with the local fluid velocity. We employ an AREPO-type approach, whichrapidly rebuilds a new high quality mesh rearranging the element shapes andneighbors in order to guarantee a robust mesh evolution even for vortex flowsand very long simulation times. The old and new Voronoi elements associated tothe same generator are connected to construct closed space--time controlvolumes, whose bottom and top faces may be polygons with a different number ofsides. We also incorporate degenerate space--time sliver elements, needed tofill the space--time holes that arise because of topology changes. The finalALE FV-DG scheme is obtained by a redesign of the fully discrete direct ALEschemes of Boscheri and Dumbser, extended here to moving Voronoi meshes andspace--time sliver elements. Our new numerical scheme is based on theintegration over arbitrary shaped closed space--time control volumes combinedwith a fully-discrete space--time conservation formulation of the governing PDEsystem. In this way the discrete solution is conservative and satisfies the GCLby construction.  Numerical convergence studies as well as a large set of benchmarks forhydrodynamics and magnetohydrodynamics (MHD) demonstrate the accuracy androbustness of the proposed method. Our numerical results clearly show that thenew combination of very high order schemes with regenerated meshes withtopology changes lead to substantial improvements compared to direct ALEmethods on conforming meshes.

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