  In this paper we continue our work on adaptive timestep control for weakly
non- stationary problems. The core of the method is a space-time splitting of
adjoint error representations for target functionals due to S\"uli and
Hartmann. The main new ingredients are (i) the extension from scalar, 1D,
conservation laws to the 2D Euler equations of gas dynamics, (ii) the
derivation of boundary conditions for a new formulation of the adjoint problem
and (iii) the coupling of the adaptive time-stepping with spatial adaptation.
For the spatial adaptation, we use a multiresolution-based strategy developed
by M\"uller, and we combine this with an implicit time discretization. The
combined space-time adaptive method provides an efficient choice of timesteps
for implicit computations of weakly nonstationary flows. The timestep will be
very large in time intervalls of stationary flow, and becomes small when a
perturbation enters the flow field. The efficiency of the solver is
investigated by means of an unsteady inviscid 2D flow over a bump.
