一种带二阶差商的波动方程新本征正交分解方法
A New Proper Orthogonal Decomposition Method with Second Difference Quotients for the Wave Equation
AI总结：
本文将差商POD方法推广至二阶差商情形，提出仅用一个快照和一个差商的新POD方法，消除了数据冗余，并证明了阻尼波动方程的逐点及能量降阶模型误差界。
摘要：
近期，研究人员探讨了偏微分方程的本征正交分解（POD）降阶模型中POD、差商（DQs）与逐点时间误差界之间的关系。在近期一项工作（Eskew和Singler，Adv. Comput. Math., 49, 2023, no. 2, Paper No. 13）中，提出了一种新的DQ POD方法，它比标准DQ POD方法计算效率更高，同时保留了标准方法保证的逐点时间误差界。在本工作中，我们将这种新的DQ POD方法推广到二阶差商（DDQs）的情形。具体而言，提出了一种利用DDQs且仅使用一个快照和一个DQ的新POD方法，并用于证明阻尼波动方程的ROM误差界。这种新方法消除了使用所有快照、DQs和DDQs的标准DDQ POD方法中的数据冗余。我们证明这种新的DDQ方法也具有类似于DQ POD的逐点时间数据误差界，并利用它证明了逐点和能量ROM误差界。我们提供了POD误差和ROM误差的数值结果以验证理论结果。我们还探讨了POD在训练区间之外模拟ROM以收集标准POD方法和DDQ POD方法快照数据的应用。
英文摘要：
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.
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