  The elucidation of many physical problems in science and engineering is
subject to the accurate numerical modelling of complex wave propagation
phenomena. Over the last decades, high-order numerical approximation for
partial differential equations has become a well-established tool. Here we
propose and study numerically the implicit approximation in time of wave
equations by a Galerkin--collocation approach that relies on a higher order
space-time finite element approach. The 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. For the fully discrete
solution, higher order regularity in time is further ensured which can be
advantageous in the discretization of multi-physics systems. The accuracy and
efficiency of the variational collocation approach is carefully studied by
numerical experiments.
