A Newton--Galerkin Method for Fluid Flow Exhibiting Uncertain Periodic Dynamics

M. Schick, V. Heuveline, O. P. Le Ma

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


The determination of stable limit-cycles plays an important role in quantifying the characteristics of dynamical systems. In practice, exact knowledge of model parameters is rarely available leading to parameter uncertainties, which can be modeled as an input of random variables. This has the effect that the limit-cycles become stochastic themselves, resulting in almost surely time-periodic solutions with a stochastic period. In this paper we introduce a novel numerical method for the computation of stable stochastic limit-cycles based on the spectral stochastic finite element method using polynomial chaos (PC). We are able to overcome the difficulties of PC regarding its well-known convergence breakdown for long term integration. To this end, we introduce a stochastic time scaling which treats the stochastic period as an additional random variable and controls the phase-drift of the stochastic trajectories, keeping the necessary PC order low. Based on the rescaled governing equations, we aim at determining an initial condition and a period such that the trajectories close after completion of one stochastic cycle. Furthermore, we verify the numerical method by computation of a vortex shedding of a flow around a circular domain with stochastic inflow boundary conditions as a benchmark problem. The results are verified by comparison to purely deterministic reference problems and demonstrate high accuracy up to machine precision in capturing the stochastic variations of the limit-cycle.
Original languageEnglish (US)
Pages (from-to)153-173
Number of pages21
JournalSIAM/ASA Journal on Uncertainty Quantification
Issue number1
StatePublished - Jan 2014
Externally publishedYes


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