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

TL;DR

This paper introduces two-scale exponential integrators for the highly oscillatory non-relativistic Klein-Gordon equation, achieving high accuracy and long-term energy conservation even as the small parameter approaches zero.

Contribution

The paper develops and analyzes new two-scale integrators with uniform accuracy and energy conservation for the Klein-Gordon equation in the nonrelativistic limit.

Findings

Achieves uniform accuracy of order three and four in time.

Proves long-time near energy conservation of the integrators.

Constructs practical integrators using symmetric and stiff order conditions.

Abstract

In this paper, we are concerned with two-scale integrators for the non-relativistic Klein--Gordon (NRKG) equation with a dimensionless parameter $0<\varepsilon\ll 1$, which is inversely proportional to the speed of light. The highly oscillatory property in time of this model corresponds to the parameter $\varepsilon$ and the equation {in the form of $\partial_{tt}u -\frac{\Delta}{\eps^2} u +\frac{1}{\eps^4}u +\frac{\lambda}{\varepsilon^2} f(u)=0$} {has a factor $1/\varepsilon^2$ in front of the nonlinearity which means that this part becomes strong when $\varepsilon$ is small. These} two aspects bring significantly numerical burdens in designing numerical methods. {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 $\mathcal{O}(h^3)$ and $\mathcal{O}(h^4)$ 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.