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Algorithm for Stabilizing a POD-Based Dynamical SystemThis algorithm provides a new way to improve the accuracy and asymptotic behavior of a low-dimensional system based on the proper orthogonal decomposition (POD). Given a data set representing the evolution of a system of partial differential equations (PDEs), such as the Navier-Stokes equations for incompressible flow, one may obtain a low-dimensional model in the form of ordinary differential equations (ODEs) that should model the dynamics of the flow. Temporal sampling of the direct numerical simulation of the PDEs produces a spatial time series. The POD extracts the temporal and spatial eigenfunctions of this data set. Truncated to retain only the most energetic modes followed by Galerkin projection of these modes onto the PDEs obtains a dynamical system of ordinary differential equations for the time-dependent behavior of the flow. In practice, the steps leading to this system of ODEs entail numerically computing first-order derivatives of the mean data field and the eigenfunctions, and the computation of many inner products. This is far from a perfect process, and often results in the lack of long-term stability of the system and incorrect asymptotic behavior of the model. This algorithm describes a new stabilization method that utilizes the temporal eigenfunctions to derive correction terms for the coefficients of the dynamical system to significantly reduce these errors.
Document ID
20100009662
Document Type
Other - NASA Tech Brief
Authors
Kalb, Virginia L. (NASA Goddard Space Flight Center Greenbelt, MD, United States)
Date Acquired
August 25, 2013
Publication Date
March 1, 2010
Publication Information
Publication: NASA Tech Briefs, March 2010
Subject Category
Man/System Technology and Life Support
Report/Patent Number
GSC-15129-1
Distribution Limits
Public
Copyright
Public Use Permitted.

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