Implicit-explicit time discretization for oseen's equation at high reynolds number with application to fractional step methods

In this paper we consider the application of implicit-explicit (IMEX) time discretizations for the incompressible Oseen equations. The pressure velocity coupling and the viscous terms are treated implicitly, while the convection term is treated explicitly. Both the second-order backward differentiation and the Crank--Nicolson methods are considered for time discretization, resulting in a scheme similar to Gear's method on the one hand and to the Adams--Bashforth of second order on the other. For the discretization in space, we consider finite element methods with stabilization on the gradient jumps. The stabilizing terms ensures inf-sup stability for equal-order interpolation and robustness at a high Reynolds number. Under suitable Courant conditions we prove stability of Gear's scheme in this regime. The stabilization allows us to prove error estimates of order O(hk+ 1 2 +\tau 2). Here h is the mesh parameter, k is the polynomial order, and \tau the time step. Finally we show that for inviscid flow (or underresolved viscous flow) the IMEX scheme can be written as a fractional step method in which only a mass matrix is inverted for each velocity component and a Poisson-type equation is solved for the pressure.