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Stability, convergence, and geometric properties of second-order-in-time space-time discretizations for linear and semilinear wave equations

Matteo Ferrari, Ilaria Perugia, Enrico Zampa

TL;DR

This work analyzes second-order-in-time space--time Galerkin discretizations for linear and semilinear wave equations by establishing precise equivalences with first-order-in-time formulations, enabling transfer of well-known stability, convergence, and energy properties. A stabilized second-order formulation is shown to be equivalent to the classical first-order DG--CG method, yielding unconditional stability and reduced unknowns, with efficient time-stepping enabled by a subquadrature interpretation. The framework is extended to semilinear problems, preserving stability and convergence, and two symplectic variants based on Gauss--Legendre and Gauss--Lobatto quadratures are proposed, corresponding to Runge--Kutta collocation schemes in time. Numerical experiments confirm energy conservation in the linear case, demonstrate convergence for the sine-Gordon equation, and illustrate the geometric structure preserved by the symplectic time integrators.

Abstract

We revisit second-order-in-time space-time discretizations of the linear and semilinear wave equations by establishing precise equivalences with first-order-in-time formulations. Focusing on schemes using continuous piecewise-polynomial trial functions in time, we analyze their stability, convergence, and geometric properties. We consider first a weak space-time formulation with test functions projected onto discontinuous polynomials of one degree lower in time, showing that it is equivalent to the scheme proposed in [French, Peterson 1996] in the linear case, and extended in [Karakashian, Makridakis 2005] to the semilinear case. In particular, this equivalence shows that this method conserves energy at mesh nodes but is not symplectic. We then introduce two symplectic variants, obtained through Gauss-Legendre and Gauss-Lobatto quadratures in time, and show that they correspond to specific Runge-Kutta time integrators. These connections clarify the geometric structure of the space-time methods considered.

Stability, convergence, and geometric properties of second-order-in-time space-time discretizations for linear and semilinear wave equations

TL;DR

This work analyzes second-order-in-time space--time Galerkin discretizations for linear and semilinear wave equations by establishing precise equivalences with first-order-in-time formulations, enabling transfer of well-known stability, convergence, and energy properties. A stabilized second-order formulation is shown to be equivalent to the classical first-order DG--CG method, yielding unconditional stability and reduced unknowns, with efficient time-stepping enabled by a subquadrature interpretation. The framework is extended to semilinear problems, preserving stability and convergence, and two symplectic variants based on Gauss--Legendre and Gauss--Lobatto quadratures are proposed, corresponding to Runge--Kutta collocation schemes in time. Numerical experiments confirm energy conservation in the linear case, demonstrate convergence for the sine-Gordon equation, and illustrate the geometric structure preserved by the symplectic time integrators.

Abstract

We revisit second-order-in-time space-time discretizations of the linear and semilinear wave equations by establishing precise equivalences with first-order-in-time formulations. Focusing on schemes using continuous piecewise-polynomial trial functions in time, we analyze their stability, convergence, and geometric properties. We consider first a weak space-time formulation with test functions projected onto discontinuous polynomials of one degree lower in time, showing that it is equivalent to the scheme proposed in [French, Peterson 1996] in the linear case, and extended in [Karakashian, Makridakis 2005] to the semilinear case. In particular, this equivalence shows that this method conserves energy at mesh nodes but is not symplectic. We then introduce two symplectic variants, obtained through Gauss-Legendre and Gauss-Lobatto quadratures in time, and show that they correspond to specific Runge-Kutta time integrators. These connections clarify the geometric structure of the space-time methods considered.
Paper Structure (14 sections, 8 theorems, 89 equations, 2 figures, 1 table)

This paper contains 14 sections, 8 theorems, 89 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The linear case
  3. The linear wave equation
  4. Discretization setup
  5. Discrete formulation
  6. Equivalence result
  7. Theoretical properties
  8. Numerical results
  9. Extension to the semilinear case
  10. Semilinear wave equations
  11. Discrete formulations
  12. Numerical results
  13. Symplectic variants based on quadrature in time
  14. First-order-in-time DG--CG time discretizations are Gauss--Legendre Runge--Kutta methods

Key Result

Proposition 2.2

For all $u_{h_t}^{p_t}, v_{h_t}^{p_t} \in S_{h_t}^{p_t}(0,T)$, it holds true that where $\{x_j^{(i)}\}_{j=1}^{p_t}$ and $\{\omega_j^{(i)}\}_{j=1}^{p_t}$ are nodes and weights of Gauss--Legendre quadrature with $p_t$ points in $[t_{i-1},t_i]$.

Figures (2)

  • Figure 1: Evolution of the discrete energy error $|E_{\boldsymbol{h}}^{p_t}(t_j)-E_{\boldsymbol{h}}^{p_t}(0)|$ using polynomial degrees $p_t=p_x=1$ (left) and $p_t=p_x=2$ (right).
  • Figure 2: Absolute errors with respect to the exact solution \ref{['eq:33a']} for varying mesh sizes $h_t=h_x$, using polynomial degrees $p_t=p_x=1$ (left) and $p_t=p_x=2$ (right).

Theorems & Definitions (26)

  • Remark 2.1: Unstabilized scheme and stabilized scheme of Zank2021
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • Remark 2.6
  • ...and 16 more