Semi - Lagrangian multistep exponential integrators for index 2 differential algebraic system

Implicit-explicit (IMEX) multistep methods are very useful for the time discretiza-tion of convection diffusion PDE problems such as the Burgers equations and also the incompressible Navier-Stokes equations. Semi-discretization in space of the latter typically gives rise to an index 2 differential-algebraic (DAE) system of equations. Runge-Kutta (RK) methods have been considered for the time discretization of such DAE systems. However, due to their implicit nature, they generally have a drawback over the IMEX multistep methods in terms of computational costs per step. In this paper we propose an exponential integration method for index 2 DAEs of a special class that includes the type arising from the incompressible Navier-Stokes problem. The methods are based on the backward differentiation formulae (BDF), belong to the class of IMEX multistep methods and are unconditionally stable.