A filtered finite difference method for a highly oscillatory nonlinear Klein--Gordon equation
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.
Cite
@article{arxiv.2504.19359,
title = {A filtered finite difference method for a highly oscillatory nonlinear Klein--Gordon equation},
author = {Yanyan Shi and Christian Lubich},
journal= {arXiv preprint arXiv:2504.19359},
year = {2026}
}
Comments
This article has been withdrawn because we have developed a substantially revised methodology and extended analysis, resulting in a new paper. The updated version is available as arXiv:2602.03322, entitled "Weighted finite difference methods for a nonlinear Klein-Gordon equation with high oscillations in space and time."