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 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 AREPOtype approach [1], 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 computational 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 need 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 t n and time t n+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 [2,3], 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 spacetime 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 over the existing state of the art in direct ALE methods.Keywords: Arbitrary-Lagrangian-Eulerian (ALE) Finite Volume (FV) and Discontinuous Galerkin (DG) schemes, arbitrary high order in space and time, moving Voronoi tessellations with topology change, a posteriori sub-cell finite volume limiter, fully-discrete one-step ADER approach for hyperbolic PDE, compressible Euler and MHD equations directly integrated by means of a high order fully discrete one-step ADER method. To the best knowledge of the authors, this is the first time that arbitrary high order accurate direct ALE FV and DG schemes are developed with an embedded mesh generator that builds a new mesh with a different topology at each time step. State of the artLagrangian algorithms [4,5,6,7,8,9,10,11,12] are characterized by a moving computational mesh displaced with a velocity chosen as close as possible to the local fluid velocity. In the Lagrangian description of the fluid, the ...

High Order ADER Schemes for Continuum Mechanics

“…Space-time predictor for sliver space-time elements When a topology change occurs, some space-time sliver elements, as those shown on the right side of Figure 8, are originated (see [90]), and the predictor procedure over them needs particular care. The problem Figure 8: Space time connectivity with topology changes and sliver element.…”

“…Left: at time t n the polygons P n 2 and P n 3 are neighbors and share the highlighted edge, instead at time t n+1 they do not touch each other; the opposite situation occurs for polygons P n 1 and P n 4 . This change of topology causes the appearance of degenerate elements of different types (refer to [90] for all the details). In particular, so-called space-time sliver elements (right) need to be taken into account when considering the space-time framework, so the predictor and the corrector step have to be a adapted to their special features.…”

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