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

Mauro Bonafini, Davide Torlo, Elena Gaburro

TL;DR

The paper provides a rigorous von Neumann stability analysis for explicit and implicit ALE-ADER-DG schemes solving hyperbolic PDEs on classical and degenerate spacetime geometries. It develops the predictor-corrector ALE framework with space-time control volumes and demonstrates that stability is governed by a CFL-based amplification matrix A(CFL, theta) (and its degenerate variants), with explicit methods requiring CFL-dependent bounds and implicit methods exhibiting unconditional stability within tested ranges. Importantly, introducing degenerate, hole-like sliver elements preserves stability and the nominal order of accuracy, showing the same CFL constraints apply as in classical geometries. These results offer a solid theoretical foundation for using degenerate elements to connect moving meshes in high-order direct ALE-DG solvers, with implications for spacetime cut-cell approaches in higher dimensions.

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.

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

TL;DR

The paper provides a rigorous von Neumann stability analysis for explicit and implicit ALE-ADER-DG schemes solving hyperbolic PDEs on classical and degenerate spacetime geometries. It develops the predictor-corrector ALE framework with space-time control volumes and demonstrates that stability is governed by a CFL-based amplification matrix A(CFL, theta) (and its degenerate variants), with explicit methods requiring CFL-dependent bounds and implicit methods exhibiting unconditional stability within tested ranges. Importantly, introducing degenerate, hole-like sliver elements preserves stability and the nominal order of accuracy, showing the same CFL constraints apply as in classical geometries. These results offer a solid theoretical foundation for using degenerate elements to connect moving meshes in high-order direct ALE-DG solvers, with implications for spacetime cut-cell approaches in higher dimensions.

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.
Paper Structure (18 sections, 1 theorem, 48 equations, 8 figures)

This paper contains 18 sections, 1 theorem, 48 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Aim and structure of this paper
  3. Explicit and implicit formulation of ALE ADER-DG methods
  4. Explicit ALE ADER-DG method
  5. Predictor step
  6. Corrector step
  7. Implicit ALE ADER-DG method
  8. Stability of ALE ADER-DG methods on classical geometries
  9. Stability of the explicit method
  10. Numerical consistency order of the explicit method
  11. Stability of the implicit method
  12. Numerical consistency order on classical geometries
  13. ALE ADER-DG methods on degenerate spacetime geometries
  14. Explicit ALE ADER-DG on degenerate geometries
  15. Implicit ALE ADER-DG on degenerate geometries
  16. ...and 3 more sections

Key Result

Theorem 4.2

On a linear advection equation $\partial_t u + \partial_x u =0$, the implicit ADER-DG method on a degenerate geometry with $\delta \leq \frac{1}{2} \min\left\{ \frac{\Delta t}{\Delta x}, 1\right\}$ is $L^2$ stable.

Figures (8)

  • Figure 1: Prototype configuration of a classical space-time domain discretization.
  • Figure 1: Semi-logarithmic plot of the function $\textrm{CFL} \mapsto \max_{\theta} \rho(A(\textrm{CFL}, \theta)) - 1$, which guarantees stability of explicit ADER-DG whenever it vanishes, see \ref{['eq.stabilitycondition']}.
  • Figure 1: Spacetime connectivity with topology changes in $2$d (left), induced hole-like sliver element (right) and neighbouring classical control volumes (middle).
  • Figure 2: Logarithmic plot of the function $\textrm{CFL} \mapsto \max_{\theta} \rho(A(\textrm{CFL}, \theta)) - 1$.
  • Figure 2: Prototype configuration of a space-time domain discretization with a degenerate (or sliver) element.
  • ...and 3 more figures

Theorems & Definitions (4)

  • Remark 3.1
  • Remark 4.1
  • Theorem 4.2: Stability for implicit ADER-DG with slivers
  • Proof 1