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Weighted finite difference methods for a nonlinear Klein--Gordon equation with high oscillations in space and time

Yanyan Shi, Christian Lubich

TL;DR

This work addresses the numerical approximation of the nonlinear Klein–Gordon equation in the nonrelativistic limit with highly oscillatory data. It develops exponentially weighted finite difference schemes in co-moving coordinates to decouple and accurately resolve counter-propagating wave packets, yielding second-order accuracy with time steps and mesh sizes that are not tied to the small parameter $\varepsilon$, and proves uniform convergence for $0<\varepsilon\le1$. The authors establish a modulated Fourier expansion for the numerical solution, deriving error bounds $O(\tau^2+ h^2+\varepsilon)$ for general initial data and $O(\tau^2+ h^2+\varepsilon^2)$ for polarized data, with the explicit method subjected to a CFL condition and the Crank–Nicolson method unconditionally stable. Consistency and stability analyses in the Wiener algebra underpin the results, and numerical experiments corroborate the theory, including polarized cases and higher oscillation regimes, demonstrating practical robustness and potential extensions to higher dimensions.

Abstract

We consider a nonlinear Klein--Gordon equation in the nonrelativistic limit regime with initial data in the form of a modulated highly oscillatory exponential. In this regime of a small scaling parameter $\varepsilon$, the solution exhibits rapid oscillations in both time and space, posing challenges for numerical approximation. We propose an explicit and an implicit exponentially weighted finite difference method. While the explicit weighted leapfrog method needs to satisfy a CFL-type stability condition, the implicit weighted Crank--Nicolson method is unconditionally stable. Both methods achieve second-order accuracy with time steps and mesh sizes that are not restricted in magnitude by $\varepsilon$. The methods are uniformly convergent in the range from arbitrarily small to moderately bounded $\varepsilon$. Numerical experiments illustrate the theoretical results.

Weighted finite difference methods for a nonlinear Klein--Gordon equation with high oscillations in space and time

TL;DR

This work addresses the numerical approximation of the nonlinear Klein–Gordon equation in the nonrelativistic limit with highly oscillatory data. It develops exponentially weighted finite difference schemes in co-moving coordinates to decouple and accurately resolve counter-propagating wave packets, yielding second-order accuracy with time steps and mesh sizes that are not tied to the small parameter , and proves uniform convergence for . The authors establish a modulated Fourier expansion for the numerical solution, deriving error bounds for general initial data and for polarized data, with the explicit method subjected to a CFL condition and the Crank–Nicolson method unconditionally stable. Consistency and stability analyses in the Wiener algebra underpin the results, and numerical experiments corroborate the theory, including polarized cases and higher oscillation regimes, demonstrating practical robustness and potential extensions to higher dimensions.

Abstract

We consider a nonlinear Klein--Gordon equation in the nonrelativistic limit regime with initial data in the form of a modulated highly oscillatory exponential. In this regime of a small scaling parameter , the solution exhibits rapid oscillations in both time and space, posing challenges for numerical approximation. We propose an explicit and an implicit exponentially weighted finite difference method. While the explicit weighted leapfrog method needs to satisfy a CFL-type stability condition, the implicit weighted Crank--Nicolson method is unconditionally stable. Both methods achieve second-order accuracy with time steps and mesh sizes that are not restricted in magnitude by . The methods are uniformly convergent in the range from arbitrarily small to moderately bounded . Numerical experiments illustrate the theoretical results.
Paper Structure (19 sections, 10 theorems, 97 equations, 2 figures)

This paper contains 19 sections, 10 theorems, 97 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preparation: Dominant terms and co-moving coordinates
  3. Solution approximation for general initial data
  4. Solution approximation for polarized initial data
  5. Reformulation in co-moving coordinates
  6. Weighted finite difference methods
  7. Exponentially weighted finite differences
  8. Exponentially weighted leapfrog algorithm
  9. Exponentially weighted Crank--Nicolson algorithm
  10. Main results and numerical experiments
  11. Results for general initial data
  12. Polarized initial data
  13. Numerical experiments
  14. Consistency
  15. Defect bound in the maximum norm
  16. ...and 4 more sections

Key Result

proposition thmcounterproposition

Let $u(t,x)$ be the solution of equation eq:KG for initial data eq:init with $0<\varepsilon\ll 1$, $\kappa\ne 0$ and smooth profile functions $a_0$ and $b_0$. Then there exists $c$ independent of $\varepsilon$ such that where $\widetilde{u}$ has the form with frequency ${\omega = \sqrt{1 + \kappa^2}}$, group velocity $c_g = \partial_\kappa \omega = \kappa/\omega<1$, and with co-moving coordinate

Figures (2)

  • Figure 1: General highly oscillatory initial data \ref{['eq:init']}. Left: error vs. $h$ for different values of $\varepsilon$. Right: error vs. $\varepsilon$ for different values of $h$. The time step is chosen according to \ref{['eq:step']}.
  • Figure 2: Polarized initial data. Left: error vs. $h$ for different values of $\varepsilon$. Right: error vs. $\varepsilon$ for different values of $h$. The time step is chosen according to \ref{['eq:step']}.

Theorems & Definitions (20)

  • proposition thmcounterproposition: Dominant terms of the solution
  • proof
  • remark thmcounterremark
  • proposition thmcounterproposition: Dominant term in the case of polarized initial data
  • proof
  • theorem 1: Dominant terms of the numerical solution
  • theorem 2: Error bound
  • remark thmcounterremark
  • theorem 3: Error bound for polarized initial data
  • lemma thmcounterlemma
  • ...and 10 more