Uniformly accurate low regularity integrators for the Klein--Gordon equation from the classical to non-relativistic limit regime

TL;DR

This paper introduces new integrators for the Klein-Gordon equation that are accurate across different regimes, including classical and highly-oscillatory non-relativistic limits, while requiring less regularity than traditional methods.

Contribution

The authors develop a novel class of integrators that are uniformly accurate across regimes and preserve the NLS limit, with improved convergence under lower regularity assumptions.

Findings

Schemes converge with order τ and τ^2 under low regularity.

Integrators accurately capture both classical and non-relativistic regimes.

Preserve the nonlinear Schrödinger limit at the discrete level.

Abstract

We propose a novel class of uniformly accurate integrators for the Klein--Gordon equation which capture classical $c=1$ as well as highly-oscillatory non-relativistic regimes $c\gg1$ and, at the same time, allow for low regularity approximations. In particular, the schemes converge with order $\tau$ and $\tau^2$, respectively, under lower regularity assumptions than classical schemes, such as splitting or exponential integrator methods, require. The new schemes in addition preserve the nonlinear Schr\"odinger (NLS) limit on the discrete level. More precisely, we will design our schemes in such a way that in the limit $c\to \infty$ they converge to a recently introduced class of low regularity integrators for NLS.