A Unified Framework for the Solution of Hyperbolic PDE Systems Using High Order Direct Arbitrary-Lagrangian–Eulerian Schemes on Moving Unstructured Meshes with Topology Change

Abstract

In this work, we review the family of direct Arbitrary-Lagrangian–Eulerian (ALE) finite vlume (FV) and discontinuous Galerkin (DG) schemes on moving meshes that at each time step are rearranged by explicitly allowing topology changes, in order to guarantee a robust mesh evolution even for high shear flow and very long evolution times. Two different techniques are presented: a local nonconforming approach for dealing with sliding lines, and a global regeneration of Voronoi tessellations for treating general unpredicted movements. Corresponding elements at consecutive times 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 nodes, with different neighbors, and even degenerate space-time sliver elements. Our final ALE FV-DG scheme is obtained by integrating, over these arbitrary shaped space-time control volumes, the space-time conservation formulation of the governing hyperbolic PDE system: so, we directly evolve the solution in time avoiding any remapping stage, being conservative and satisfying the GCL by construction. Arbitrary high order of accuracy in space and time is achieved through a fully discrete one-step predictor–corrector ADER approach, also integrated with well balancing techniques to further improve the accuracy and to maintain exactly even at discrete level many physical invariants of the studied system. A large set of different numerical tests has been carried out in order to check the accuracy and the robustness of our methods for both smooth and discontinuous problems, in particular in the case of vortical flows.