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Scalable ADER-DG Transport Method with Polynomial Order Independent CFL Limit

Kieran Ricardo, Kenneth Duru

TL;DR

The paper addresses the explicit time-step bottleneck of high-order DG methods for transport-dominated problems. It introduces a locally implicit but globally explicit ADER-DG scheme that solves element-local implicit problems, yielding a maximum stable time step governed by element width and independent of polynomial order (up to a factor of 1/√d in d dimensions). A 1D stability proof and a semi-analytical von Neumann analysis for 2D and 3D establish the method's stability limits, while numerical experiments across linear and nonlinear tests (including cubed-sphere advection) demonstrate quasi-optimal convergence and favorable CFL behavior. The approach promises enhanced scalability on HPC architectures by reducing inter-element communication while maintaining high-order accuracy for transport problems.

Abstract

Discontinuous Galerkin (DG) methods are known to suffer from increasingly restrictive explicit time-step constraints as the polynomial order increases, limiting their efficiency at high orders for explicit time-stepping schemes. In this paper, we introduce a novel \emph{locally implicit}, but \emph{globally explicit} ADER-DG scheme designed for transport-dominated problems. The method achieves a maximum stable time step governed by an element-width based CFL condition that is independent of the polynomial degree. By solving a set of element-local implicit problems at each time step, our approach more effectively utilises the domain of dependence. As a result, our method remains stable for CFL numbers up to $\approx 1/\sqrt{d}$ in $d$ spatial dimensions. We provide a rigorous stability proof in one dimension, and extend the analysis to two and three dimensions using a semi-analytical von Neumann stability analysis. The accuracy and convergence of the method are demonstrated through numerical experiments for both linear and nonlinear test cases, including numerical simulations of a transport problem on a cubed sphere 2D manifold embedded in 3D.

Scalable ADER-DG Transport Method with Polynomial Order Independent CFL Limit

TL;DR

The paper addresses the explicit time-step bottleneck of high-order DG methods for transport-dominated problems. It introduces a locally implicit but globally explicit ADER-DG scheme that solves element-local implicit problems, yielding a maximum stable time step governed by element width and independent of polynomial order (up to a factor of 1/√d in d dimensions). A 1D stability proof and a semi-analytical von Neumann analysis for 2D and 3D establish the method's stability limits, while numerical experiments across linear and nonlinear tests (including cubed-sphere advection) demonstrate quasi-optimal convergence and favorable CFL behavior. The approach promises enhanced scalability on HPC architectures by reducing inter-element communication while maintaining high-order accuracy for transport problems.

Abstract

Discontinuous Galerkin (DG) methods are known to suffer from increasingly restrictive explicit time-step constraints as the polynomial order increases, limiting their efficiency at high orders for explicit time-stepping schemes. In this paper, we introduce a novel \emph{locally implicit}, but \emph{globally explicit} ADER-DG scheme designed for transport-dominated problems. The method achieves a maximum stable time step governed by an element-width based CFL condition that is independent of the polynomial degree. By solving a set of element-local implicit problems at each time step, our approach more effectively utilises the domain of dependence. As a result, our method remains stable for CFL numbers up to in spatial dimensions. We provide a rigorous stability proof in one dimension, and extend the analysis to two and three dimensions using a semi-analytical von Neumann stability analysis. The accuracy and convergence of the method are demonstrated through numerical experiments for both linear and nonlinear test cases, including numerical simulations of a transport problem on a cubed sphere 2D manifold embedded in 3D.

Paper Structure

This paper contains 19 sections, 5 theorems, 51 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Space-time discontinuous Galerkin method
  3. Implicit space-time discontinuous Galerkin method
  4. Standard ADER discontinous Galerkin method
  5. Novel locally implicit ADER-DG method
  6. Notation
  7. 1D
  8. 2D
  9. 3D
  10. 1D Stability Analysis
  11. Analysis
  12. Stability of the eigenvalue problem
  13. Von Neumann Stability Analysis
  14. Numerical results
  15. 2D linear advection
  16. ...and 4 more sections

Key Result

Theorem 2.1

The implicit ADER-DG method eq:imp-ader with constant advection velocity vector $\boldsymbol{u}$ is unconditionally stable, that is

Figures (10)

  • Figure 1: New locally implicit ADER-DG method in two spatial dimensions. A dotted boundary indicates no numerical flux, and coloured solid boundaries denote upwind numerical fluxes calculated with the predictor of the corresponding colour.
  • Figure 2: Numerically calculated $\norm{A}_w$.
  • Figure 3: Amplification factor minus 1 for the 1D scheme.
  • Figure 4: Amplification factor minus 1 for the 2D scheme.
  • Figure 5: Amplification factor minus 1 for the 3D scheme.
  • ...and 5 more figures

Theorems & Definitions (9)

  • Theorem 2.1
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Theorem 4.3
  • proof
  • Theorem 4.4
  • proof