On the treatment of topology changes on 3D polyhedral moving meshes via 4D space-time hole-like elements in direct ALE ADER-DG methods

This work investigates a novel approach for the high order evolution of hyperbolic PDEs using ADER discontinuous Galerkin schemes within a direct Arbitrary-Lagrangian–Eulerian (ALE) framework on 3D moving polyhedral meshes with topology changes. Our direct ALE method is based on the PDE integration over 4D (3D+time) space–time control volumes connecting the elements of two subsequent tessellations, so to simultaneously evolve the solution both in time and between the two different meshes in an effective and high order manner. In this way, we also avoid any complex and expensive projection-reconstruction techniques and any mesh intersection operation typical of indirect ALE schemes. The crucial step consists in the strategy for building space–time control volumes that also connect elements with different shapes and neighborhoods due to a change in topology. In fact, simply linking existing elements by collapsing or expanding their edges would leave a ‘‘hole’’ in the space–time domain. To fill it, we introduce additional degenerate elements that we call hole-like elements. These are 4D objects with zero 3D volume at both the beginning and end of the timestep, but which possess a strictly non-zero 4D space–time volume. Given the uniqueness of this space–time approach in 3D+time and the necessity of characterizing the geometry of such elements, the main objective of this paper is the formal geometrical and numerical description of the method. Specifically, we show that despite being non-trivial to visualize, these elements are well-defined and ultimately even easy to manage. In particular, we provide here a detailed characterization of the hole-like elements needed to connect two polyhedral tessellations, corresponding to 2-3, 3-2, and 4-4 flips on the underlying Delaunay tetrahedralization. We describe how to identify and construct them, how to adapt the numerical method to their degenerate geometry by introducing for them a locally implicit treatment, and we provide new and intuitive visualization strategies. Finally, a set of benchmarks, simple in their wave structure but challenging for moving meshes, is used to prove that the method is fully conservative, satisfies the Geometric Conser- vation Law condition, and maintains the correct order of convergence even in the presence of frequent topology changes.