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.

翻译：近年来，研究者探讨了本征正交分解（POD）、差商（DQs）与偏微分方程POD降阶模型的时间逐点误差界之间的关系。近期工作（Eskew和Singler, Adv. Comput. Math., 49, 2023, 第2期, 论文编号13）提出了一种新的含DQ的POD方法，其计算效率高于标准DQ POD方法，并保留了标准方法的时间逐点误差保证。本文将该新型DQ POD方法推广至二次差商（DDQs）情形。具体而言，我们发展了一种仅使用单一快照和单一DQ的新型DDQ POD方法，并将其用于证明阻尼波动方程的ROM误差界。该新方法消除了标准DDQ POD方法中因使用全部快照、DQs和DDQs造成的数据冗余。我们证明该新型DDQ方法同样具有与DQ POD相似的时间逐点数据误差界，并据此证明了逐点误差和能量ROM误差界。通过数值实验结果展示POD误差与ROM误差，验证理论结论。进一步探讨了POD在训练区间外模拟ROM中的应用，以收集标准POD方法和DDQ POD方法的快照数据。