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Title: Long-time uncertainty propagation using generalized polynomial chaos and flow map composition

We present an efficient and accurate method for long-time uncertainty propagation in dynamical systems. Uncertain initial conditions and parameters are both addressed. The method approximates the intermediate short-time flow maps by spectral polynomial bases, as in the generalized polynomial chaos (gPC) method, and uses flow map composition to construct the long-time flow map. In contrast to the gPC method, this approach has spectral error convergence for both short and long integration times. The short-time flow map is characterized by small stretching and folding of the associated trajectories and hence can be well represented by a relatively low-degree basis. The composition of these low-degree polynomial bases then accurately describes the uncertainty behavior for long integration times. The key to the method is that the degree of the resulting polynomial approximation increases exponentially in the number of time intervals, while the number of polynomial coefficients either remains constant (for an autonomous system) or increases linearly in the number of time intervals (for a non-autonomous system). The findings are illustrated on several numerical examples including a nonlinear ordinary differential equation (ODE) with an uncertain initial condition, a linear ODE with an uncertain model parameter, and a two-dimensional, non-autonomous double gyre flow.
Authors:
 [1] ;  [2] ;  [3]
  1. Department of Mechanical Engineering, The Cooper Union for the Advancement of Science and Art, NY 10003 (United States)
  2. Department of Applied Mathematics, University of Washington, Seattle, WA 98195 (United States)
  3. Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544 (United States)
Publication Date:
OSTI Identifier:
22382123
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Computational Physics; Journal Volume: 274; Other Information: Copyright (c) 2014 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; APPROXIMATIONS; CHAOS THEORY; CONVERGENCE; DIFFERENTIAL EQUATIONS; ERRORS; NONLINEAR PROBLEMS; POLYNOMIALS; TRAJECTORIES; TWO-DIMENSIONAL CALCULATIONS