  The single-step one-shot method has proven to be very efficient for
PDE-constrained optimization where the partial differential equation (PDE) is
solved by an iterative fixed point solver. In this approach, the simulation and
optimization tasks are performed simultaneously in a single iteration. If the
PDE is unsteady, finding an appropriate fixed point iteration is non-trivial.
In this paper, we provide a framework that makes the single-step one-shot
method applicable for unsteady PDEs that are solved by classical time-marching
schemes. The One-shot method is applied to an optimal control problem with
unsteady incompressible Navier-Stokes equations that are solved by an industry
standard simulation code. With the Van-der-Pol oscillator as a generic model
problem, the modified simulation scheme is further improved using adaptive time
scales. Finally, numerical results for the advection-diffusion equation are
presented.
