We discuss numerical properties of continuous Galerkin-Petrov and discontinuous
Galerkin time discretizations applied to the heat equation as a prototypical example
for scalar parabolic partial differential equations. For the space discretization, we use
biquadratic quadrilateral finite elements on general two-dimensional meshes. We discuss
implementation aspects of the time discretization as well as efficient methods for solving
the resulting block systems. Here, we compare a preconditioned BiCGStab solver as a
Krylov space method with an adapted geometrical multigrid solver. Only the convergence
of the multigrid method is almost independent of the mesh size and the time step leading
to an efficient solution process. By means of numerical experiments we compare the
different time discretizations with respect to accuracy and computational costs.