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 ε = 1 \varepsilon = 1 up to long wave limit regime ε ≪ 1 \varepsilon \ll 1 on the natural time scale of the PDE t = O ( 1 ε ) t= \mathcal {O}(\frac {1}{\varepsilon }) . Most notably the global error of our new schemes is of order τ ε \tau \varepsilon (for the first-order scheme) and τ 2 ε \tau ^2 \varepsilon (for the second-order scheme) on time intervals of length O ( 1 ε ) \mathcal {O}\left ( \frac {1}{\varepsilon }\right ) .