  In this paper we consider a conservative discretization of the
two-dimensional incompressible Navier--Stokes equations. We propose an
extension of Arakawa's classical finite difference scheme for fluid flow in the
vorticity-stream function formulation to a high order discontinuous Galerkin
approximation. In addition, we show numerical simulations that demonstrate the
accuracy of the scheme and verify the conservation properties, which are
essential for long time integration. Furthermore, we discuss the massively
parallel implementation on graphic processing units.
