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Error estimates of time-splitting schemes for nonlinear Klein--Gordon equation with rough data

Lun Ji, Xiaofei Zhao

TL;DR

This work analyzes the convergence of Lie-Trotter and Strang time-splitting schemes for the nonlinear Klein-Gordon equation on the torus with rough data. It develops a two-pronged framework: Sobolev-space analysis for $s>\tfrac12$ and a discrete Bourgain-space approach for very rough data with $s\in(\tfrac{11}{40},\tfrac{51}{40}]$, including a projection $\Pi_\tau$ to control high-frequency components. The authors prove optimal temporal-error bounds in Sobolev spaces, and establish corresponding global error estimates in the Bourgain regime via discrete Bourgain multilinear estimates, complemented by nonlinear-estimate generalizations and numerical verifications. The results demonstrate sharp convergence orders for both Lie and Strang splitting under low regularity and introduce a discrete Bourgain machinery applicable to general second-order wave models, with practical implications for reliable simulations of dispersive PDEs with rough initial data.

Abstract

In this work, we consider the convergence analysis of time-splitting schemes for the nonlinear Klein--Gordon/wave equation under rough initial data. The optimal error bounds of the Lie splitting and the Strang splitting are established with sharp dependence on the regularity index of the solution from a wide range that is approaching the lower bound for well-posedness. Particularly for very rough data, the technique of discrete Bourgain space is utilized and developed, which can apply for general second-order wave models. Numerical verifications are provided.

Error estimates of time-splitting schemes for nonlinear Klein--Gordon equation with rough data

TL;DR

This work analyzes the convergence of Lie-Trotter and Strang time-splitting schemes for the nonlinear Klein-Gordon equation on the torus with rough data. It develops a two-pronged framework: Sobolev-space analysis for and a discrete Bourgain-space approach for very rough data with , including a projection to control high-frequency components. The authors prove optimal temporal-error bounds in Sobolev spaces, and establish corresponding global error estimates in the Bourgain regime via discrete Bourgain multilinear estimates, complemented by nonlinear-estimate generalizations and numerical verifications. The results demonstrate sharp convergence orders for both Lie and Strang splitting under low regularity and introduce a discrete Bourgain machinery applicable to general second-order wave models, with practical implications for reliable simulations of dispersive PDEs with rough initial data.

Abstract

In this work, we consider the convergence analysis of time-splitting schemes for the nonlinear Klein--Gordon/wave equation under rough initial data. The optimal error bounds of the Lie splitting and the Strang splitting are established with sharp dependence on the regularity index of the solution from a wide range that is approaching the lower bound for well-posedness. Particularly for very rough data, the technique of discrete Bourgain space is utilized and developed, which can apply for general second-order wave models. Numerical verifications are provided.

Paper Structure

This paper contains 15 sections, 15 theorems, 101 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Time-splitting methods
  3. Error estimates in Sobolev space
  4. Cauchy problem for (\ref{['kgb']})
  5. Local error estimate
  6. Proof of Theorem \ref{['theosob']}
  7. Error estimates in Bourgain space
  8. A Bourgain framework
  9. Local error estimates
  10. Proof of Theorem \ref{['theobourg']}
  11. Nonlinear estimates and generalizations
  12. Proof of Lemma \ref{['theomultsob']}
  13. Proof of Theorem \ref{['theomultbourg']}
  14. Generalization
  15. Numerical Experiment

Key Result

Theorem 3.1

For $s>\frac{1}{2}$ and the initial data $z_0(x)\in H^s(\mathbb{T}^2),~z_1(x)\in H^{s-1}(\mathbb{T}^2)$, let $z$ be the exact solution to kg on $[0,T]$. Moreover, we denote $u=z+i\langle\nabla\rangle^{-1}z_t$ (i.e., $u$ is the solution to the equivalent equation kgb). Furthermore, let $u_n,~v_n$ be where $C_T$ depends on $T$ and $\tau_0$ but is independent of $n$ and $\tau$.

Figures (4)

  • Figure 1: $H^{\frac{1}{2}}$ temporal error of the Lie splitting scheme for the two-dimensional NKG with rough initial data $z_0+\langle\nabla\rangle^{-1}z_1\in H^s$. Left: $s=0.6$; right: $s=5/6$.
  • Figure 2: $H^{\frac{1}{2}}$ temporal error of the Strang splitting scheme for the two-dimensional NKG with rough initial data $z_0+\langle\nabla\rangle^{-1}z_1\in H^s$. Left: $s=2/3$; right: $s=1.5$.
  • Figure 3: $L^2$ temporal error of the Strang splitting scheme for the one-dimensional NKG with rough initial data $z_0+\langle\nabla\rangle^{-1}z_1\in H^s$. Left: $s=1/6$; right: $s=2/3$.
  • Figure 4: $H^\frac{11}{40}$ temporal error of the filtered Lie and Strang splitting schemes for the two-dimensional NKG with very rough initial data $z_0+\langle\nabla\rangle^{-1}z_1\in H^s$. Left: $s=0.4$; right: $s=0.5$.

Theorems & Definitions (30)

  • Theorem 3.1: Convergence in Sobolev
  • Lemma 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • proof
  • Remark 3.5
  • Theorem 4.1: Convergence in Bourgain
  • Lemma 4.2
  • ...and 20 more