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arXiv · 2602.04988

A uniformly accurate multiscale time integrator for the nonlinear Klein-Gordon equation in the nonrelativistic regime via simplified transmission conditions

Weizhu Bao·Caoyi Liu
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Abstract

We propose a new and simplified multiscale time integrator Fourier pseudospectral (MTI-FP) method for the nonlinear Klein-Gordon equation (NKGE) with a dimensionless parameter epsilon in (0,1] inversely proportional to the speed of light, and establish its uniform first-order accuracy in time in the nonrelativistic regime, i.e. 0 < epsilon << 1. In this regime, the solution of the NKGE is highly oscillatory in time with O(epsilon^2)-wavelength, which brings significant difficulties in designing uniformly accurate numerical methods. The MTI-FP is based on (i) a multiscale decomposition by frequency of the NKGE in each time interval with simplified transmission conditions, and (ii) an exponential wave integrator for temporal discretization and a Fourier pseudospectral method for spatial discretization. By adapting the energy method and the mathematical induction, we obtain two error bounds in H1-norm at O(h^{m0}+tau^2/epsilon^2) and O(h^{m0}+tau+epsilon^2) with mesh size h, time step tau and m0 an integer dependent on the regularity of the solution, which immediately implies a uniformly accurate error bound O(h^{m0}+tau) with respect to epsilon in (0,1]. In addition, by adopting a linear interpolation of the micro variables with the multiscale decomposition in each time interval, we obtain a uniformly accurate numerical solution for any time t larger than zero. Thus the proposed MTI-FP method has a super resolution property in time in terms of the Shannon sampling theory, i.e. accurate numerical solutions can be obtained even when the time step is much bigger than the O(epsilon^2)-wavelength. Extensive numerical results are reported to confirm our error bounds and demonstrate their super resolution in time. Finally the proposed MTI-FP method is applied to study numerically convergence rates of the NKGE to its different limiting models in the nonrelativistic regime.

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Weizhu Bao, Caoyi Liu. 2026-02-04. A uniformly accurate multiscale time integrator for the nonlinear Klein-Gordon equation in the nonrelativistic regime via simplified transmission conditions. https://arxiv.org/abs/2602.04988

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