Abstract
We propose a class of semi-Lagrangian methods of high approximation order in space and time, based on spectral element space discretizations and exponential integrators of Runge-Kutta type. The methods were presented in Celledoni and Kometa (J Sci Comput 41(1):139-164, 2009) for simpler convection-diffusion equations. We discuss the extension of these methods to the Navier-Stokes equations, and their implementation using projections. Semi-Lagrangian methods up to order three are implemented and tested on various examples. The good performance of the methods for convection-dominated problems is demonstrated with numerical experiments.