Two-scale integrators with high accuracy and long-time conservations for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime

¹ School of Mathematics and Statistics, Xi'an Jiaotong University, 710049 Xi'an, P.R. China
² School of Mathematics and Statistics, Northwestern Polytechnical University, 710072 Xi'an, P.R. China

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 17 April 2025
Accepted: 3 January 2026

Abstract

In this paper, we are concerned with two-scale integrators for the non-relativistic Klein-Gordon (NRKG) equation with a dimensionless parameter 0 < ε ≪ 1, which is inversely proportional to the speed of light. The highly oscillatory property in time of this model corresponds to the parameter ε and the equation in the form of ∂_ttu − Δ/ε²u + 1/ε⁴u + λ/ε²f(u) = 0 has a factor 1/ε² in front of the nonlinearity which means that this part becomes strong when ε is small. We propose a class of two-scale integrators which is constructed based on some reformulations to the system, Fourier pseudo-spectral method and exponential integrators. Two practical integrators up to order three and four are constructed by using some symmetric conditions and the stiff order conditions of implicit exponential integrators. The convergence of the obtained integrators is rigorously studied, and it is shown that the uniform accuracy in time is 𝒪(h³) and 𝒪(h⁴) for the time stepsize h. The near energy conservation over long times is also established for the multi-stage integrators by using modulated Fourier expansions. Numerical results on a NRKG equation show that the proposed integrators have high accuracy, excellent long time energy conservation and competitive efficiency.