  Proper orthogonal decomposition methods for model reduction utilize
information about the solution at certain time and parameter points to generate
a reduced space basis. In this paper, we compare two proper orthogonal
decomposition methods for reducing large systems of ODEs. The first method is
based on collecting snapshots from the solutions only; the second method uses
snapshots from both the solutions and their time derivatives. To compare the
methods, we derive new bounds for the 2-norm of the approximation error induced
by the each of the methods. The bounds are represented as a sum of two terms:
the first depends on the size of the first neglected singular value while the
second depends only on the spacings between the snapshots. We performed
numerical experiments to compare the errors from the two model reduction
methods applied to the semidiscretized FitzHugh-Nagumo system and investigated
the relation between the behavior of the numerically observed error and the
error bounds. We find that the error bounds, though not tight, provide insights
and justification for using time derivative snapshots in POD model reduction
for dynamical systems.
