# Comparison of numerical methods for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime

## TL;DR

This paper systematically compares various numerical methods for solving the nonlinear Klein-Gordon equation in the nonrelativistic limit, focusing on efficiency, accuracy, and scalability as the parameter approaches zero.

## Contribution

It provides a comprehensive comparison of multiple numerical methods for the NKGE, highlighting their performance and $ ext{ε}$-resolution capabilities in the nonrelativistic limit regime.

## Key findings

- Multiscale time integrator shows superior $ ext{ε}$-resolution.
- Different methods exhibit varying efficiency and accuracy in the nonrelativistic limit.
- Convergence rates of the NKGE to limiting models are analyzed.

## Abstract

Different efficient and accurate numerical methods have recently been proposed and analyzed for the nonlinear Klein-Gordon equation (NKGE) with a dimensionless parameter $\varepsilon\in (0,1]$, which is inversely proportional to the speed of light. In the nonrelativestic limit regime, i.e. $0<\varepsilon\ll1$, the solution of the NKGE propagates waves with wavelength at $O(1)$ and $O(\varepsilon^2)$ in space and time, respectively, which brings significantly numerical burdens in designing numerical methods. We compare systematically spatial/temporal efficiency and accuracy as well as $\varepsilon$-resolution (or $\varepsilon$-scalability) of different numerical methods including finite difference time domain methods, time-splitting method, exponential wave integrator, limit integrator, multiscale time integrator, two-scale formulation method and iterative exponential integrator. Finally, we adopt the multiscale time integrator to study the convergence rates from the NKGE to its limiting models when $\varepsilon\to0^+$.