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A filtered finite difference method for a highly oscillatory nonlinear Klein--Gordon equation

Yanyan Shi, Christian Lubich

TL;DR

This work tackles the numerical approximation of highly oscillatory nonlinear Klein–Gordon equations in the nonrelativistic limit using a filtered finite difference method in co-moving coordinates. By analyzing a modulated-Fourier expansion and enforcing a consistency condition between the time step, mesh size, and the small parameter $\varepsilon$, the authors obtain a second-order, uniformly accurate scheme that remains stable as $\varepsilon$ varies from $0$ to $1$, and can handle large $\tau$ and $h$ when $h\gg\varepsilon$ and $\tau\gg\varepsilon^2$. The analysis combines defect bounds in the Wiener algebra with both linear and nonlinear stability arguments to establish an overall error of $O(\tau^2+h^2+\varepsilon)$, and the method naturally separates counterpropagating waves via co-moving frames. Numerical experiments corroborate the theoretical rates, show asymptotic-preserving behavior, and demonstrate robustness across regimes. The approach provides a practical, provably accurate tool for simulating rapidly oscillatory dispersive waves in nonlinear Klein–Gordon-type systems.

Abstract

We consider a nonlinear Klein--Gordon equation in the nonrelativistic limit regime with highly oscillatory initial data in the form of a modulated plane wave. In this regime, the solution exhibits rapid oscillations in both time and space, posing challenges for numerical approximation. We propose a filtered finite difference method that achieves second-order accuracy with time steps and mesh sizes that are not restricted in magnitude by the small parameter. Moreover, the method is uniformly convergent in the range from arbitrarily small to moderately bounded scaling parameters. Numerical experiments illustrate the theoretical results.

A filtered finite difference method for a highly oscillatory nonlinear Klein--Gordon equation

TL;DR

This work tackles the numerical approximation of highly oscillatory nonlinear Klein–Gordon equations in the nonrelativistic limit using a filtered finite difference method in co-moving coordinates. By analyzing a modulated-Fourier expansion and enforcing a consistency condition between the time step, mesh size, and the small parameter , the authors obtain a second-order, uniformly accurate scheme that remains stable as varies from to , and can handle large and when and . The analysis combines defect bounds in the Wiener algebra with both linear and nonlinear stability arguments to establish an overall error of , and the method naturally separates counterpropagating waves via co-moving frames. Numerical experiments corroborate the theoretical rates, show asymptotic-preserving behavior, and demonstrate robustness across regimes. The approach provides a practical, provably accurate tool for simulating rapidly oscillatory dispersive waves in nonlinear Klein–Gordon-type systems.

Abstract

We consider a nonlinear Klein--Gordon equation in the nonrelativistic limit regime with highly oscillatory initial data in the form of a modulated plane wave. In this regime, the solution exhibits rapid oscillations in both time and space, posing challenges for numerical approximation. We propose a filtered finite difference method that achieves second-order accuracy with time steps and mesh sizes that are not restricted in magnitude by the small parameter. Moreover, the method is uniformly convergent in the range from arbitrarily small to moderately bounded scaling parameters. Numerical experiments illustrate the theoretical results.
Paper Structure (10 sections, 7 theorems, 68 equations, 2 figures, 1 algorithm)

This paper contains 10 sections, 7 theorems, 68 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Dominant term of the exact solution
  3. Numerical method and main results
  4. Consistency
  5. Defect bound in the maximum norm
  6. Defect bound in the Wiener algebra norm
  7. Stability
  8. Linear stability analysis in the Wiener algebra
  9. Nonlinear stability
  10. Numerical experiments

Key Result

proposition thmcounterproposition

There exists a positive constant $c$ such that where $u_{\text{app}}$ has the form with $\xi^{\pm} = x \mp c_g t/\varepsilon$, frequency $\omega = \sqrt{1 + \kappa^2}$, and the group velocity $c_g = \partial_\kappa \omega = \kappa/\omega <1$. The functions $a^{+}(t, \xi^+)$ and $a^{-}(t, \xi^-)$ satisfy the following nonlinear Schrödinger equations:

Figures (2)

  • Figure 1: Error vs. $h$ with different $\varepsilon$ for filtered finite difference method.
  • Figure 2: Error vs. $\varepsilon$ with different $h$ for filtered finite difference method.

Theorems & Definitions (17)

  • proposition thmcounterproposition
  • proof
  • remark thmcounterremark
  • remark thmcounterremark
  • remark thmcounterremark
  • theorem 1: Dominant term of the numerical solution
  • remark thmcounterremark
  • theorem 2: Error bound
  • remark thmcounterremark
  • lemma 2
  • ...and 7 more