Stability analysis of Arbitrary-Lagrangian-Eulerian ADER-DG methods on classical and degenerate spacetime geometries

Abstract

In this paper, we present a thorough von Neumann stability analysis of explicit and implicit Arbitrary-Lagrangian-Eulerian (ALE) ADER discontinuous Galerkin (DG) methods on classical and degenerate spacetime geometries for hyperbolic equations. First, we rigorously study CFL stability conditions for the explicit ADER-DG method, confirming results widely used in the literature while specifying their limitations. Moreover, we discuss stability bounds for ALE methods and characterize the admissible range of grid velocities once a target CFL is fixed. Next, we extend the stability study to ADER-DG in the presence of degenerate spacetime elements, with zero size at the beginning and the end of the time step, but with a non zero spacetime volume. This kind of elements has been introduced in a series of articles on direct ALE methods by Gaburro et al. to connect via spacetime control volumes regenerated Voronoi tessellations after a topology change. Here, we imitate this behavior in a 1d surrogate setting by fictitiously inserting degenerate elements in between two cells. We show that over this simplified degenerate spacetime geometry, both for the explicit and implicit ADER-DG, the von Neumann analysis leads to the same CFL stability conditions as those for classical geometries, laying the theoretical foundations for their use in the context of ALE methods.