In this paper, we extend our work for the heat equation [1] and for the Stokes equations [2]
to the nonstationary Navier-Stokes equations. We present fully implicit continuous Galerkin-
Petrov (cGP) and discontinuous Galerkin (dG) time stepping schemes for incompressible flow
problems which are, in contrast to standard approaches like for instance the Crank-Nicolson
scheme, of higher order in time. In particular, we implement and analyze numerically the
higher order dG(1) and cGP(2)-methods which are super-convergent of 3rd, resp., 4th order
in time, while for the space discretization, the well-known LBB-stable finite element pair
Q2/Pdisc
1 is used. The discretized systems of nonlinear equations are treated by using the
Newton method, and the associated linear subproblems are solved by means of a monolithic
multigrid method with a blockwise Vanka-like smoother [3]. We perform nonstationary simu-
lations for two benchmarking configurations to analyze the temporal accuracy and efficiency
of the presented time discretization schemes. As a first test problem, we consider a classical
flow around cylinder benchmark [4]. Here, we concentrate on the nonstationary behavior
of the flow patterns with periodic oscillations and examine the ability of the different time
discretization schemes to capture the dynamics of the flow. As a second test case, we consider
the nonstationary flow through a Venturi pipe [5, 6]. The objective of this simulation is to
control the instantaneous and mean flux through this device.