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Discontinuous Galerkin time integration for second-order differential problems: formulations, analysis, and analogies

Gabriele Ciaramella, Martin J. Gander, Ilario Mazzieri

TL;DR

This work analyzes discontinuous Galerkin methods as time integrators for linear second-order oscillatory ODEs, presenting two formulations: a direct second-order DG (DG2) and a first-order DG (DG1). It establishes convergence and equivalence results, showing that DG2 is connected to Newmark schemes and GLMs, while DG1 aligns with Lobatto IIIC implicit Runge–Kutta methods. Through detailed algebraic formulations and spectral analysis across polynomial degrees $r=1,2,3$, the authors derive precise order and stability properties, including scenarios where DG1 outperforms DG2 in accuracy and conditioning. Numerical experiments corroborate the theoretical findings, indicating that DG1 is typically preferable due to higher accuracy and better conditioning, while DG2 offers a rigorous GLM interpretation and high-order potential. These results provide a comprehensive framework for choosing DG time-stepping strategies in space-time discretizations of wave-like problems.

Abstract

We thoroughly investigate Discontinuous Galerkin (DG) discretizations as time integrators for second-order oscillatory systems, considering both second-order and first-order formulations of the original problem. Key contributions include new convergence analyses for the second-order formulation and equivalence proofs between DG and classical time-stepping schemes (such as Newmark schemes and general linear methods). In addition, the chapter provides a detailed review and convergence analysis for the first-order formulation, alongside comparisons of the proposed schemes in terms of accuracy, consistency, and computational cost.

Discontinuous Galerkin time integration for second-order differential problems: formulations, analysis, and analogies

TL;DR

This work analyzes discontinuous Galerkin methods as time integrators for linear second-order oscillatory ODEs, presenting two formulations: a direct second-order DG (DG2) and a first-order DG (DG1). It establishes convergence and equivalence results, showing that DG2 is connected to Newmark schemes and GLMs, while DG1 aligns with Lobatto IIIC implicit Runge–Kutta methods. Through detailed algebraic formulations and spectral analysis across polynomial degrees , the authors derive precise order and stability properties, including scenarios where DG1 outperforms DG2 in accuracy and conditioning. Numerical experiments corroborate the theoretical findings, indicating that DG1 is typically preferable due to higher accuracy and better conditioning, while DG2 offers a rigorous GLM interpretation and high-order potential. These results provide a comprehensive framework for choosing DG time-stepping strategies in space-time discretizations of wave-like problems.

Abstract

We thoroughly investigate Discontinuous Galerkin (DG) discretizations as time integrators for second-order oscillatory systems, considering both second-order and first-order formulations of the original problem. Key contributions include new convergence analyses for the second-order formulation and equivalence proofs between DG and classical time-stepping schemes (such as Newmark schemes and general linear methods). In addition, the chapter provides a detailed review and convergence analysis for the first-order formulation, alongside comparisons of the proposed schemes in terms of accuracy, consistency, and computational cost.
Paper Structure (19 sections, 14 theorems, 159 equations, 6 figures)

This paper contains 19 sections, 14 theorems, 159 equations, 6 figures.

Table of Contents

  1. Introduction
  2. A brief review of some integrators
  3. The Newmark method
  4. Implicit Runge-Kutta methods
  5. General linear methods
  6. Time DG methods for second-order equations: DG2
  7. Algebraic formulation for the DG2 discretization
  8. Polynomial degree r=1
  9. Polynomial degree r=2
  10. Polynomial degree r=3
  11. DG2 as GLMs
  12. Time DG methods for first-order systems: DG1
  13. Algebraic formulation for DG discretization in first-order form: DG1
  14. DG1 as Implicit Runge-Kutta scheme
  15. Polynomial degree r=1
  16. ...and 4 more sections

Key Result

Lemma 1

Consider the scheme eq:error for $n=0,\dots,N-1$. Let $G=WDW^{-1}$ be the eigen-decomposition of $G$, with $D$ denoting the diagonal matrix whose entries are the eigenvalues of $G$. Assume that the spectral radius of $G$ satisfies $\rho(G)\leq 1$. If $\bm e_0 = \bm 0$, $W=O(\Delta t^q)$, and $W^{-1}

Figures (6)

  • Figure 1: Example of a time domain partition and a zoom with the values $t_n^+$ and $t_n^-$.
  • Figure 2: Spectral radius of $G$ as a function of $\sqrt{\lambda}\Delta t$ for different values of $a$.
  • Figure 3: DG2: computed convergence errors $\| \bm e_n \|$ as a function of the time step $\Delta t$ for $r=1,2,3$ with $s=0$ (left) and $s=\frac{\Delta t^2}{2}$ (right).
  • Figure 4: DG2: computed consistency errors $\| V^{-1} \bm \theta_n \|$ as a function of the time step $\Delta t$ for $r=1,2,3$ with $s=0$ (left) and $s=\frac{\Delta t^2}{2}$ (right).
  • Figure 5: DG1: computed convergence errors $\| \bm e_n \|$ (left) and $| \bm e_{N-1}|$ (right) as a function of the time step $\Delta t$ for $r=1,2,3$.
  • ...and 1 more figures

Theorems & Definitions (29)

  • Example 1: Newmark as a GLM
  • Example 2: Order conditions of Newmark as a GLM
  • Lemma 1: accuracy and consistency
  • proof
  • Theorem 1: $\mathbb{P}^1$-DG is a Newmark scheme
  • proof
  • Lemma 2: Eigen-decomposition of $G$ for $\mathbb{P}^1$
  • proof
  • Theorem 2: Consistency and accuracy for $\mathbb{P}^1$
  • proof
  • ...and 19 more