Semi-Lagrangian multistep exponential integrators for index 2 differential–algebraic systems

Implicit-explicit (IMEX) multistep methods are very useful for the time discretization of convection diffusion PDE problems such as the Burgers equations and the incompressible Navier–Stokes equations. In the latter as well as in PDE models of plasma physics and of electromechanical systems, semi-discretization in space gives rise to differential–algebraic (DAE) system of equations often of index higher than 1. In this paper we propose a new class of exponential integrators for index 2 DAEs arising from the semi-discretization of PDEs with a dominating and typically nonlinear convection term. This class of problems includes the incompressible Navier–Stokes equations. The integration methods are based on the backward differentiation formulae (BDF) and they can be applied without modifications in the semi-Lagrangian integration of convection diffusion problems. The approach gives improved performance at low viscosity regimes.