  We consider a control constrained parabolic optimal control problem and use
variational discretization for its time semi-discretization. The state equation
is treated with a Petrov-Galerkin scheme using a piecewise constant Ansatz for
the state and piecewise linear, continuous test functions. This results in
variants of the Crank-Nicolson scheme for the state and the adjoint state.
Exploiting a superconvergence result we prove second order convergence in time
of the error in the controls. Moreover, the piecewise linear and continuous
parabolic projection of the discrete state on the dual time grid provides a
second order convergent approximation of the optimal state without further
numerical effort. Numerical experiments confirm our analytical findings.
