Time-dependent generalized polynomial chaos

Authors:
Marc Gerritsma;Jan-Bart van der Steen;Peter Vos;George Karniadakis
Affiliations:
Department of Aerospace Engineering, TU Delft, The Netherlands;Siemens Nederland N.V., Prinses Beatrixlaan 800 , P.O. Box 16068, 2500 BB The Hague, Netherlands;Flemish Institute for Technological Research (VITO), Unit Environmental Modelling, Boeretang 200, 2400 Mol, Belgium;Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
Venue:
Journal of Computational Physics
Year:
2010

Citing 12
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Abstract

Generalized polynomial chaos (gPC) has non-uniform convergence and tends to break down for long-time integration. The reason is that the probability density distribution (PDF) of the solution evolves as a function of time. The set of orthogonal polynomials associated with the initial distribution will therefore not be optimal at later times, thus causing the reduced efficiency of the method for long-time integration. Adaptation of the set of orthogonal polynomials with respect to the changing PDF removes the error with respect to long-time integration. In this method new stochastic variables and orthogonal polynomials are constructed as time progresses. In the new stochastic variable the solution can be represented exactly by linear functions. This allows the method to use only low order polynomial approximations with high accuracy. The method is illustrated with a simple decay model for which an analytic solution is available and subsequently applied to the three mode Kraichnan-Orszag problem with favorable results.