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Almost conservation of the harmonic actions for fully discretized nonlinear Klein--Gordon equations at low regularity

Charbella Abou Khalil, Joackim Bernier

TL;DR

This work treats the nonlinear Klein–Gordon equation on the circle as a nearly integrable Hamiltonian system and proves that, under a CFL-compatible fully discretized scheme with a mollified impulse time integrator, the low harmonic actions remain almost conserved for long times when the initial data are small in the energy space. The authors develop a backward error analysis to construct a modified Hamiltonian $H_h$ that closely tracks the numerical flow and then put $H_h$ into a partial Birkhoff normal form, facilitated by a strong non-resonance property of the discretized frequencies. From this, they derive almost invariance of the modified harmonic actions and, via a priori energy control, almost preservation of the low actions for times $nh\le\varepsilon^{-r}$. The approach extends prior high-regularity results to low regularity, aligns with numerical observations, and offers a robust framework potentially applicable to other dispersive PDE discretizations.

Abstract

Close to the origin, the nonlinear Klein--Gordon equations on the circle are nearly integrable Hamiltonian systems which have infinitely many almost conserved quantities called harmonic actions or super-actions. We prove that, at low regularity and with a CFL number of size 1, this property is preserved if we discretize the nonlinear Klein--Gordon equations with the symplectic mollified impulse methods. This extends previous results of D. Cohen, E. Hairer and C. Lubich to non-smooth solutions.

Almost conservation of the harmonic actions for fully discretized nonlinear Klein--Gordon equations at low regularity

TL;DR

This work treats the nonlinear Klein–Gordon equation on the circle as a nearly integrable Hamiltonian system and proves that, under a CFL-compatible fully discretized scheme with a mollified impulse time integrator, the low harmonic actions remain almost conserved for long times when the initial data are small in the energy space. The authors develop a backward error analysis to construct a modified Hamiltonian that closely tracks the numerical flow and then put into a partial Birkhoff normal form, facilitated by a strong non-resonance property of the discretized frequencies. From this, they derive almost invariance of the modified harmonic actions and, via a priori energy control, almost preservation of the low actions for times . The approach extends prior high-regularity results to low regularity, aligns with numerical observations, and offers a robust framework potentially applicable to other dispersive PDE discretizations.

Abstract

Close to the origin, the nonlinear Klein--Gordon equations on the circle are nearly integrable Hamiltonian systems which have infinitely many almost conserved quantities called harmonic actions or super-actions. We prove that, at low regularity and with a CFL number of size 1, this property is preserved if we discretize the nonlinear Klein--Gordon equations with the symplectic mollified impulse methods. This extends previous results of D. Cohen, E. Hairer and C. Lubich to non-smooth solutions.
Paper Structure (24 sections, 25 theorems, 252 equations, 1 figure)

This paper contains 24 sections, 25 theorems, 252 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Context and motivation
  3. Discretization
  4. Main result
  5. Discussions about the proof
  6. Numerical experiments
  7. Hamiltonian formalism
  8. Complex variables and associated functional setting
  9. Class of Hamiltonian functions
  10. Vector field and Poisson bracket estimates
  11. Flows
  12. Backward error analysis
  13. Formal derivation of the system of cohomological equations
  14. The cohomological equations
  15. The modified Hamiltonian
  16. ...and 9 more sections

Key Result

Theorem 1.1

For all $r\geq 1$, there exists $\beta_r \geq 1$, such that for almost all $\rho >0$, all $\delta>0$, all $K\geq 1$ and all $h>0$ satisfying there exists $\varepsilon_0(r,\rho)>0$ depending only on $r$ and $\rho$ such that if $(q^n,p^n)_{n\geq 0}$ is a sequence of real-valued functions on $\mathbb{T}_K$ solution of the fully discretized nonlinear Klein--Gordon equation eq:KG_fd and whose initial

Figures (1)

  • Figure 1: Evolution of the logarithm of the super actions $\log_{10} J_k(q^n,p^n)$ for different regularity exponents $s_0$ and index ranges $k$ and for times $hn\leq T=4000$.

Theorems & Definitions (58)

  • Theorem 1.1
  • Definition 2.1: Poisson bracket
  • Remark 2.2
  • Definition 2.3: Adjoint representation $\mathrm{ad}$
  • Definition 2.4: Symplectic Map
  • Definition 2.5: Momentum $\mathcal{M}(j,\sigma)$
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • proof
  • ...and 48 more