  We show that the discrete operator stemming from the time and space
discretization of evolutionary partial differential equations can be
represented in terms of a single Sylvester matrix equation. A novel solution
strategy that combines projection techniques with the full exploitation of the
entry-wise structure of the involved coefficient matrices is proposed. The
resulting scheme is able to efficiently solve problems with a tremendous number
of degrees of freedom while maintaining a low storage demand as illustrated in
several numerical examples.
