Uniformly Accurate Low Regularity Integrators for the Klein--Gordon Equation from the Classical to NonRelativistic Limit Regime

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