  We consider a family of variational time discretizations that are
generalizations of discontinuous Galerkin (dG) and continuous Galerkin-Petrov
(cGP) methods. The family is characterized by two parameters. One describes the
polynomial ansatz order while the other one is associated with the global
smoothness that is ensured by higher order collocation conditions at both ends
of the subintervals. The presented methods provide the same stability
properties as dG or cGP. Provided that suitable quadrature rules of Hermite
type for evaluating the integrals in the variational conditions are used, the
variational time discretization methods are connected to special collocation
methods. For this case, we will present error estimates, numerical experiments,
and a computationally cheap postprocessing that allows to increase both the
accuracy and the global smoothness by one order.
