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

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.

💡 Research Summary

This paper provides a comprehensive von Neumann stability analysis of both explicit and implicit Arbitrary‑Lagrangian‑Eulerian (ALE) ADER‑DG (Arbitrary high‑order DERivative Discontinuous Galerkin) schemes applied to hyperbolic conservation laws. The authors first set up the ADER‑DG method in a direct ALE framework for a scalar nonlinear hyperbolic PDE in one space dimension, defining spatial elements, spacetime control volumes, and three families of basis functions (spatial, spacetime, and moving). The explicit method consists of a predictor step, where a local spacetime polynomial qⁿᵢ is obtained by a fixed‑point Picard iteration, and a corrector step, where an ALE Riemann solver (Russo‑type flux) provides numerical fluxes across element faces. The implicit method simultaneously solves for all spacetime polynomials using the same basis but treats the whole system globally, allowing larger time steps.

For the explicit scheme, the authors linearize the method around a constant state, insert a Fourier mode e^{iκx}, and derive an amplification matrix G(κ,Δt,Δx,v) that depends on the mesh velocity v, the physical wave speed a, and the Courant number λ = aΔt/Δx. By analyzing the spectral radius of G, they recover the classic CFL limit λ_max ≈ 1/(2N+1) (N is the polynomial degree) and explicitly show that stability requires the mesh velocity to satisfy |v| < a·CFL. If the mesh moves faster than the physical wave, the scheme becomes unstable regardless of the time step.

The novel contribution concerns degenerate spacetime elements—cells whose spatial size collapses to zero at the beginning and end of a time step but retain a non‑zero spacetime volume. These elements are used in recent direct ALE methods to connect meshes after topology changes. The authors mimic this situation in 1‑D by inserting fictitious degenerate elements between two regular cells. Performing the same von Neumann analysis, they find that the amplification matrix is unchanged by the presence of degenerate geometry; consequently the CFL constraint remains identical to the classical case. This result holds for both explicit and implicit formulations, indicating that the ADER‑DG predictor‑corrector structure inherently compensates for the geometric singularity.

Numerical experiments on linear and nonlinear test problems confirm the theoretical predictions: the observed stability limits match the derived CFL numbers, and the insertion of degenerate elements does not affect convergence rates or error growth. For the implicit scheme, an L‑stability analysis shows unconditional stability for linear problems, while for nonlinear problems the authors note that robust Newton‑Krylov solvers and appropriate preconditioners are essential to retain the enlarged time‑step advantage.

In summary, the paper (i) rigorously derives CFL conditions for explicit ALE ADER‑DG, (ii) quantifies the influence of mesh velocity on stability, and (iii) proves that degenerate spacetime geometries do not alter these conditions. These findings provide a solid theoretical foundation for using ADER‑DG in complex ALE simulations involving mesh adaptation, topology changes, and moving boundaries. Future work suggested includes extending the analysis to multiple dimensions, systems of equations, and fully nonlinear L‑stability proofs.