  Recently, researchers have investigated the relationship between proper
orthogonal decomposition (POD), difference quotients (DQs), and pointwise in
time error bounds for POD reduced order models of partial differential
equations. In a recent work (Eskew and Singler, Adv. Comput. Math., 49, 2023,
no. 2, Paper No. 13), a new approach to POD with DQs was developed that is more
computationally efficient than the standard DQ POD approach and it also retains
the guaranteed pointwise in time error bounds of the standard method. In this
work, we extend this new DQ POD approach to the case of second difference
quotients (DDQs). Specifically, a new POD method utilizing DDQs and only one
snapshot and one DQ is developed and used to prove ROM error bounds for the
damped wave equation. This new approach eliminates data redundancy in the
standard DDQ POD approach that uses all of the snapshots, DQs, and DDQs. We
show that this new DDQ approach also has pointwise in time data error bounds
similar to DQ POD and use it to prove pointwise and energy ROM error bounds. We
provide numerical results for the POD errors and ROM errors to demonstrate the
theoretical results. We also explore an application of POD to simulating ROMs
past the training interval for collecting the snapshot data for the standard
POD approach and the DDQ POD method.
