  We analyze, from the viewpoint of positivity preservation, certain
discretizations of a fundamental partial differential equation, the
one-dimensional advection equation with periodic boundary condition. The full
discretization is obtained by coupling a finite difference spatial
semi-discretization (the second- and some higher-order centered difference
schemes, or the Fourier spectral collocation method) with an arbitrary
$\theta$-method in time (including the forward and backward Euler methods, and
a second-order method by choosing $\theta\in [0,1]$ suitably). The full
discretization generates a two-parameter family of circulant matrices
$M\in\mathbb{R}^{m\times m}$, where each matrix entry is a rational function in
$\theta$ and $\nu$. Here, $\nu$ denotes the CFL number, being proportional to
the ratio between the temporal and spatial discretization step sizes. The
entrywise non-negativity of the matrix $M$ -- which is equivalent to the
positivity preservation of the fully discrete scheme -- is investigated via
discrete Fourier analysis and also by solving some low-order parametric linear
recursions. We find that positivity preservation of the fully discrete system
is impossible if the number of spatial grid points $m$ is even. However, it
turns out that positivity preservation of the fully discrete system is
recovered for \emph{odd} values of $m$ provided that $\theta\ge 1/2$ and $\nu$
are chosen suitably. These results are interesting since the systems of
ordinary differential equations obtained via the spatial semi-discretizations
studied are \emph{not} positivity preserving.
