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Explicit symmetric low-regularity integrators for the semilinear Klein-Gordon equation

Zhirui Shen, Bin Wang

TL;DR

This paper addresses numerical approximation of the nonlinear Klein-Gordon equation on the torus under low regularity. It introduces an explicit symmetric low-regularity framework and derives two schemes, sLRI1 (first-order) and sLRI2 (second-order), by symmetrizing a base exponential integrator and incorporating a low-regularity correction. The main contributions are rigorous convergence results in energy space under weakened smoothness assumptions and demonstrated near-energy conservation over long times, supported by numerical experiments showing improved accuracy and efficiency relative to non-symmetric LRIs. The approach promises robust long-time behavior for nonlinear wave simulations and is adaptable to other nonlinear wave equations.

Abstract

This paper is concerned with the design and analysis of symmetric low-regularity integrators for the semilinear Klein-Gordon equation. We first propose a general symmetrization procedure that allows for the systematic construction of symmetric schemes from existing explicit (non-symmetric) integrators. Applying this procedure, we derive two novel schemes. Error analyses show that both integrators achieve their optimal convergence orders in the energy space under significantly relaxed regularity assumptions. Furthermore, the symmetry property ensures that the convergence order of a first-order symmetric scheme improves as the regularity of the exact solution increases. A numerical experiment demonstrates that the proposed second-order symmetric scheme nearly preserves the system energy over extended periods.

Explicit symmetric low-regularity integrators for the semilinear Klein-Gordon equation

TL;DR

This paper addresses numerical approximation of the nonlinear Klein-Gordon equation on the torus under low regularity. It introduces an explicit symmetric low-regularity framework and derives two schemes, sLRI1 (first-order) and sLRI2 (second-order), by symmetrizing a base exponential integrator and incorporating a low-regularity correction. The main contributions are rigorous convergence results in energy space under weakened smoothness assumptions and demonstrated near-energy conservation over long times, supported by numerical experiments showing improved accuracy and efficiency relative to non-symmetric LRIs. The approach promises robust long-time behavior for nonlinear wave simulations and is adaptable to other nonlinear wave equations.

Abstract

This paper is concerned with the design and analysis of symmetric low-regularity integrators for the semilinear Klein-Gordon equation. We first propose a general symmetrization procedure that allows for the systematic construction of symmetric schemes from existing explicit (non-symmetric) integrators. Applying this procedure, we derive two novel schemes. Error analyses show that both integrators achieve their optimal convergence orders in the energy space under significantly relaxed regularity assumptions. Furthermore, the symmetry property ensures that the convergence order of a first-order symmetric scheme improves as the regularity of the exact solution increases. A numerical experiment demonstrates that the proposed second-order symmetric scheme nearly preserves the system energy over extended periods.
Paper Structure (9 sections, 6 theorems, 86 equations, 10 figures)

This paper contains 9 sections, 6 theorems, 86 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Explicit symmetric low-regularity integrators
  3. Construction of a specific explicit symmetric LRI
  4. General method for constructing symmetric scheme
  5. Convergence results
  6. Convergence analysis for sLRI1
  7. Convergence analysis for sLRI2
  8. Numerical experiments
  9. Conclusion

Key Result

Lemma 3.1

Suppose that $r\in(0,1)$ and $W\in H^{1+r}(\Bbb T)\times H^{r}(\Bbb T)$. Then it holds that

Figures (10)

  • Figure 1: Global errors $\mathrm{err}(1)$ of the the differential schemes for the NLSE (\ref{['numerical']}) with initial data in $H^{\theta}(\Bbb T)$ of regularity $\theta=1.0$ and $\theta=1.5$ .
  • Figure 2: Global errors $\mathrm{err}(1)$ of the differential schemes for the NLSE (\ref{['numerical']}) with smooth initial data of regularity $\theta=2$ and $\theta=10$ .
  • Figure 3: The convergence orders of sLRI1 under regularity regimes $\theta=1$ and $\theta=1.2$ .
  • Figure 4: The convergence orders of sLRI1 under regularity regimes $\theta=1.4$ and $\theta=1.6$ .
  • Figure 5: The convergence orders of sLRI1 under regularity regimes $\theta=1.8$ and $\theta=2$ .
  • ...and 5 more figures

Theorems & Definitions (14)

  • Definition 2.1
  • Remark 2.1
  • Lemma 3.1
  • Lemma 3.2: Kato-Ponce
  • Proposition 3.1
  • proof
  • Remark 3.1
  • Theorem 3.1
  • proof
  • Remark 3.2
  • ...and 4 more