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Solving nonlinear Klein-Gordon equation with non-smooth solution by a geometric low-regularity integrator

Bin Wang, Zhen Miao, Yaolin Jiang

TL;DR

This work develops a simple geometric low-regularity integrator for the nonlinear Klein-Gordon equation in dimensions $d=1,2,3$, built from a two-step symmetric trigonometric scheme. It achieves second-order accuracy in the energy space under weak regularity $H^{1+\frac{d}{4}}\times H^{\frac{d}{4}}$, while preserving time symmetry to enable rigorous long-time analysis via modulated Fourier expansions. The authors prove local and global error bounds, and establish long-time near-conservation of discrete energy, actions, and momentum, by constructing a modulated Fourier expansion and almost-invariants, then patching short-time results to long times. Numerical tests corroborate the theoretical advantages, showing efficient computations (a single nonlinear evaluation per step) and superior long-time energy behavior compared with existing LR methods. Overall, the scheme offers a computationally light, structure-preserving approach for NKGE with reduced regularity requirements and strong long-time performance guarantees.

Abstract

In this paper, we formulate and analyse a geometric low-regularity integrator for solving the nonlinear Klein-Gordon equation in the $d$-dimensional space with $d=1,2,3$. The integrator is constructed based on the two-step trigonometric method and thus it has a simple form. Error estimates are rigorously presented to show that the integrator can achieve second-order time accuracy in the energy space under the regularity requirement in $H^{1+\frac{d}{4}}\times H^{\frac{d}{4}}$. Moreover, the time symmetry of the scheme ensures its good long-time energy, momentum and action conservations which are rigorously proved by the technique of modulated Fourier expansions. A numerical test is presented and the numerical results demonstrate the superiorities of the new integrator over some existing methods.

Solving nonlinear Klein-Gordon equation with non-smooth solution by a geometric low-regularity integrator

TL;DR

This work develops a simple geometric low-regularity integrator for the nonlinear Klein-Gordon equation in dimensions , built from a two-step symmetric trigonometric scheme. It achieves second-order accuracy in the energy space under weak regularity , while preserving time symmetry to enable rigorous long-time analysis via modulated Fourier expansions. The authors prove local and global error bounds, and establish long-time near-conservation of discrete energy, actions, and momentum, by constructing a modulated Fourier expansion and almost-invariants, then patching short-time results to long times. Numerical tests corroborate the theoretical advantages, showing efficient computations (a single nonlinear evaluation per step) and superior long-time energy behavior compared with existing LR methods. Overall, the scheme offers a computationally light, structure-preserving approach for NKGE with reduced regularity requirements and strong long-time performance guarantees.

Abstract

In this paper, we formulate and analyse a geometric low-regularity integrator for solving the nonlinear Klein-Gordon equation in the -dimensional space with . The integrator is constructed based on the two-step trigonometric method and thus it has a simple form. Error estimates are rigorously presented to show that the integrator can achieve second-order time accuracy in the energy space under the regularity requirement in . Moreover, the time symmetry of the scheme ensures its good long-time energy, momentum and action conservations which are rigorously proved by the technique of modulated Fourier expansions. A numerical test is presented and the numerical results demonstrate the superiorities of the new integrator over some existing methods.
Paper Structure (13 sections, 4 theorems, 117 equations, 2 figures)

This paper contains 13 sections, 4 theorems, 117 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Geometric low-regularity integrator and numerical test
  3. Explicit symmetric low-regularity integrator
  4. Numerical test
  5. Convergence
  6. Estimate of the local errors $\zeta_n^{u}, \zeta_n^{v}$
  7. Proof of global errors
  8. Long term analysis
  9. Main result of this section
  10. Proof of Theorem \ref{['main theo1']}
  11. The result of modulated Fourier expansion
  12. Almost-invariants
  13. Proof of long time conservations

Key Result

Theorem 3.1

(Error estimate) For the nonlinear function $f$ of the Klein--Gordon equation klein-gordon, the Lipschitz continuity conditions $\left\vert f^{(k)}(w)\right\vert\leq C_1$ are assumed for $w\in \mathbb{R}$ and $k=1,2,3$. Under the regularity condition $(u(0,x), \partial_tu(0,x))\in [H^{1+\frac{d}{4}} where $C$ is a positive constant which is independent of the stepsize $h$ and $n$ but depends on $C

Figures (2)

  • Figure 1: Errors of the numerical solutions with initial data $H^{\theta}(\mathbb{T}) \times H^{\theta-1}(\mathbb{T})$ against $h$ (1st row) and CPU time (2nd row) with $h =1/2^k$, where $k=2,3,\ldots,9$.
  • Figure 2: Errors of the energy with initial data $H^{\theta}(\mathbb{T}) \times H^{\theta-1}(\mathbb{T})$ against time $t$ for different $\theta$ and $h$.

Theorems & Definitions (11)

  • Definition 2.1
  • Remark 2.2
  • Theorem 3.1
  • proof
  • Definition 4.1
  • Theorem 4.3
  • Lemma 4.4
  • proof
  • Lemma 4.5
  • proof
  • ...and 1 more