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

TL;DR

This paper conducts a comprehensive stability analysis of ALE ADER-DG methods on classical and degenerate spacetime geometries, establishing stability conditions and extending results to complex degenerate elements.

Contribution

It provides the first detailed von Neumann stability analysis of ALE ADER-DG methods on degenerate spacetime geometries, confirming stability constraints and extending applicability.

Findings

CFL stability conditions are confirmed for explicit ALE ADER-DG methods.

Stability constraints remain unchanged even with degenerate spacetime elements.

Theoretical foundation for using ALE ADER-DG methods with degenerate geometries is established.

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 the CFL stability conditions for the explicit ADER-DG method, confirming results widely used in the literature while specifying their limitations. Moreover, we highlight under which conditions on the mesh velocity the ALE methods, constrained to a given CFL, are actually stable. 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 1d by fictitiously inserting degenerate elements in between two cells. Then, we show that over this degenerate spacetime geometry, both for the explicit and implicit ADER-DG, the CFL stability constraints remain the same as those for classical geometries, laying the theoretical foundations for their use in the context of ALE methods.