Parareal Time-Stepping for Limit-Cycle Computation of the Incompressible Navier-Stokes Equations with Uncertain Periodic Dynamics

The computation of limit-cycles in time periodic flow problems plays a crucial role in quantifying its dynamical characteristics. Since in many applications model parameters are often subject to uncertainty, the limit-cycle becomes a stochastic quantity itself. In this work we introduce two types of shooting methods based on Polynomial Chaos and the stochastic Galerkin projection. Polynomial Chaos is known to exhibit a convergence breakdown in time, which our proposed algorithms are able to overcome. The first algorithm is a re-interpretation of a Newton-Galerkin method as a single shooting approach. The second one extends this idea using the parareal algorithm on a time domain decomposition. It uses a fine grid propagator, which can be computed in parallel and a coarse grid propagator defined by a low order Polynomial Chaos expansion. We evaluate the convergence properties on a suitable benchmark problem (time periodic vortex shedding) showing promising results in capturing accurately the periodic dynamics of the flow.

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