A New Proper Orthogonal Decomposition Method with Second Difference Quotients for the Wave Equation
Abstract
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 ROM errors to demonstrate the theoretical results. We also explore an application of POD to simulate ROMs past the training interval for collecting the snapshot data for the standard POD approach and the DDQ POD method.
Recommended Citation
A. Janes and J. R. Singler, "A New Proper Orthogonal Decomposition Method with Second Difference Quotients for the Wave Equation," Journal of Computational and Applied Mathematics, vol. 457, article no. 116279, Elsevier, Mar 2025.
The definitive version is available at https://doi.org/10.1016/j.cam.2024.116279
Department(s)
Mathematics and Statistics
Keywords and Phrases
Proper orthogonal decomposition; Reduced order models; Second difference quotients; Wave equation
International Standard Serial Number (ISSN)
0377-0427
Document Type
Article - Journal
Document Version
Citation
File Type
text
Language(s)
English
Rights
© 2025 Elsevier, All rights reserved.
Publication Date
15 Mar 2025
Comments
National Science Foundation, Grant 2111421