Time integrators for dispersive equations in the long wave regime

TL;DR

This paper introduces new time integrators for dispersive equations that accurately simulate long wave regimes, maintaining convergence over extended times with rates depending on the small parameter .

Contribution

The paper presents a novel class of time integrators capable of capturing long wave dynamics with convergence rates proportional to  over extended time scales.

Findings

Achieved convergence with order   over long times

Successfully modeled solution dynamics from classical to long wave regimes

Provided schemes that work uniformly for   regimes

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 $ \varepsilon = 1$ up to long wave limit regime $ \varepsilon \ll 1 $ on the natural time scale of the PDE $t= \mathcal{O}(\frac{1}{\varepsilon})$. Most notably our new schemes converge with rates at order $\tau \varepsilon$ over long times $t= \frac{1}{\varepsilon}$.