Interdisciplinary Applied Mathematics

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Discussion on Weighted Basis versus Error in the Solution: It is important to note that weighting is a concept introduced to improve the accuracy over certain time scales or periods rather than a technique that can be used to improve the accuracy over the entire time period. In fact, reduced-order modeling using a weighted basis usually exhibits slightly higher error in the time period that is less significantly weighted. Typically, reduced-order modeling exhibits a very nonuniform error in the whole time domain, i.e., it might behave very well in certain time periods but not be able to capture the basic characteristics in certain other time periods. By using a weighted KL basis, it is possible to achieve a more uniform reduction in error in the solution.

A second question of significant interest is, how does a weighted basis compare with other bases in capturing the system transient? There is no easy answer without a detailed mathematical analysis. However, we do know that (1) increasing the number of basis functions used in approximating, for example, the velocities and pressure generally improves the accuracy of the simulation; (2) different methods generate different bases, and the number of significant basis functions that need to be included in each method is different. The accuracy of the solution is influenced by both the number of basis functions and the quality of the bases. Typically, the number of basis functions that need to be included in a weighted approach is less than the number of basis functions that need to be included in the classical KL approach for comparable accuracy.

Example: Transient Analysis of Electroosmotic Flow

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