  We introduce and analyze a class of Galerkin-collocation discretization
schemes in time for the wave equation. Its conceptual basis is the
establishment of a direct connection between the Galerkin method for the time
discretization and the classical collocation methods, with the perspective of
achieving the accuracy of the former with reduced computational costs provided
by the latter in terms of less complex linear algebraic systems. Continuously
differentiable in time discrete solutions are obtained by the application of a
special quadrature rule involving derivatives. Optimal order error estimates
are proved for fully discrete approximations based on the Galerkin-collocation
approach. Further, the concept of Galerkin-collocation approximation is
extended to twice continuously differentiable in time discrete solutions. A
direct connection between the two families by a computationally cheap
post-processing is presented. The error estimates are illustrated by numerical
experiments.
