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

Technical Summary: Stability Analysis of Arbitrary-Lagrangian-Eulerian ADER-DG Methods

Problem Statement
The Arbitrary-Lagrangian-Eulerian (ALE) framework is widely used to solve hyperbolic partial differential equations (PDEs) on moving meshes, particularly when topology changes occur. In this context, Direct ALE methods connect meshes at different time steps via spacetime control volumes. Recent developments by Gaburro et al. introduced "degenerate" spacetime elements (sliver elements) to handle topology changes where cells may have zero spatial size at the beginning or end of a time step but possess a non-zero spacetime volume. While ADER-DG (Arbitrary high-order DERivatives Discontinuous Galerkin) methods are effective for these problems, a rigorous theoretical stability analysis for these schemes—specifically on degenerate spacetime geometries—has been missing. Existing literature often relies on empirical CFL (Courant-Friedrichs-Lewy) bounds or limited stability studies for low polynomial orders, leaving a gap in understanding the stability constraints for high-order explicit and implicit ALE ADER-DG methods on both classical and degenerate geometries.

Methodology
The authors employ a von Neumann stability analysis to investigate the stability of explicit and implicit ALE ADER-DG methods. The study focuses on the linear advection equation as a proxy for hyperbolic systems.

1. Formulation: The paper first establishes the mathematical framework for both explicit and implicit ALE ADER-DG schemes.
   • Explicit Scheme: Utilizes a predictor-corrector approach. The predictor step constructs a local spacetime polynomial approximation within each control volume (or sliver element) using a fixed-point iteration. The corrector step updates the solution using moving basis functions and numerical fluxes (Rusanov-type) at interfaces.
   • Implicit Scheme: Solves for the global spacetime polynomial simultaneously across all control volumes using a Newton-GMRES iteration, directly integrating the PDE over the spacetime control volumes.
2. Degenerate Geometry Modeling: To analyze degenerate geometries, the authors introduce a 1D surrogate setting where standard interfaces between control volumes are replaced by "sliver elements." These elements mimic the hole-like slivers used in 2D/3D topology changes, having zero spatial width at $t_n$ and $t_{n+1}$ but a non-zero spacetime volume.
3. Stability Analysis:
   • Classical Geometries: The authors derive amplification matrices for the explicit and implicit schemes. They compute the spectral radius $\rho$ of these matrices over a range of CFL numbers and phase angles.
   • Degenerate Geometries: The analysis is extended to include sliver elements. The domain is modeled as a periodic block containing a standard control volume and a sliver element. An amplification matrix dependent on the CFL number, the sliver width parameter $\delta$, and the phase angle is constructed.
   • Numerical Verification: Theoretical stability bounds are validated by computing discrete amplification factors over fine grids of CFL values and polynomial degrees ($N=1$ to $9$). Consistency orders are also verified numerically.

Key Contributions and Results

• Refinement of Explicit Stability Bounds: For classical geometries, the study confirms that for low polynomial degrees ($N \le 3$), the stability limits align with widely used empirical CFL bounds. However, for higher orders ($N \ge 4$), the paper demonstrates that commonly used empirical CFL values actually violate strict von Neumann stability conditions (where $\rho > 1$). The authors identify significantly lower, rigorous CFL limits required for strict stability in high-order explicit schemes.
• Grid Velocity Constraints: The analysis provides a theoretical characterization of the admissible range of grid velocities for a fixed target CFL, showing how mesh motion affects stability constraints.
• Implicit Unconditional Stability: The von Neumann analysis and a theoretical proof (Theorem 1) confirm that the implicit ALE ADER-DG method is unconditionally stable for the linear advection equation on classical geometries, regardless of the time step size.
• Stability on Degenerate Geometries:
  • Explicit Case: The introduction of sliver elements does not degrade stability. The discrete amplification factor for the degenerate setting is found to be less than or equal to that of the classical setting. Consequently, the same CFL bounds applicable to classical geometries are valid for degenerate ones. The authors note that the implicit treatment of the sliver predictor step within the explicit global scheme may slightly increase the acceptable CFL limit.
  • Implicit Case: The implicit method remains unconditionally stable even in the presence of sliver elements, provided the sliver width parameter $\delta$ satisfies a specific geometric constraint relative to the time step and grid size (Theorem 2).
• Consistency: Numerical experiments confirm that both explicit and implicit schemes maintain their expected order of consistency ($N+1$) on both classical and degenerate geometries.

Significance and Claims
The paper claims to fill a critical theoretical gap by providing the first rigorous von Neumann stability analysis for ALE ADER-DG methods on degenerate spacetime geometries. The primary significance lies in validating the use of degenerate elements (slivers) for connecting moving meshes with topology changes. The results demonstrate that the use of these zero-size spatial elements does not impose additional stability restrictions or reduce the allowable time step compared to classical geometries. This finding provides a theoretical foundation for the practical application of direct ALE methods in complex scenarios involving topology changes and paves the way for the development of new spacetime cut cell-based methods. The authors emphasize that while empirical CFL values are often used in practice (mitigated by viscosity and limiters), the derived rigorous bounds are essential for a complete theoretical understanding of the method's behavior.