Time integrators for dispersive equations in the long wave regime

Abstract

We introduce a novel class of time integrators for dispersive equations which allow us to reproduce the dynamics of the solution from the classical $\epsilon =1$ up to long wave limit regime $\epsilon \ll 1$ on the natural time scale of the PDE $t=\mathcal{O}(\frac{1}{\epsilon })$. Most notably our new schemes converge with rates at order $\tau \epsilon$ over long times $t=\frac{1}{\epsilon }$.