Asymptotic preserving schemes for the Klein-Gordon equation in the non-relativistic limit regime

TL;DR

This paper develops asymptotic preserving numerical schemes for the Klein-Gordon equation in the non-relativistic limit, enabling stable and efficient simulations as the speed of light becomes very large.

Contribution

The authors introduce a novel asymptotic expansion approach that leads to numerical algorithms robust against large oscillations caused by high speeds.

Findings

Algorithms remain stable as c tends to infinity

Efficient simulation of Klein-Gordon in non-relativistic limit

High accuracy in capturing oscillatory behavior

Abstract

We consider the Klein-Gordon equation in the non-relativistic limit regime, i.e. the speed of light c tending to infinity. We construct an asymptotic expansion for the solution with respect to the small parameter depending on the inverse of the square of the speed of light. As the first terms of this asymptotic can easily be simulated our approach allows us to construct numerical algorithms that are robust with respect to the large parameter c producing high oscillations in the exact solution.