  We propose, analyze, and test a novel continuous data assimilation reduced
order model (DA-ROM) for simulating incompressible flows. While ROMs have a
long history of success on certain problems with recurring dominant structures,
they tend to lose accuracy on more complicated problems and over longer time
intervals. Meanwhile, continuous data assimilation (DA) has recently been used
to improve accuracy and, in particular, long time accuracy in fluid simulations
by incorporating measurement data into the simulation. This paper synthesizes
these two ideas, in an attempt to address inaccuracies in ROM by applying DA,
especially over long time intervals and when only inaccurate snapshots are
available. We prove that with a properly chosen nudging parameter, the proposed
DA-ROM algorithm converges exponentially fast in time to the true solution, up
to discretization and ROM truncation errors. Finally, we propose a strategy for
nudging adaptively in time, by adjusting dissipation arising from the nudging
term to better match true solution energy. Numerical tests confirm all results,
and show that the DA-ROM strategy with adaptive nudging can be highly effective
at providing long time accuracy in ROMs.
