Dynamically Orthogonal Numerical Schemes for Efficient Stochastic Advection and Lagrangian Transport
Author(s)
Feppon, Florian Jeremy; Lermusiaux, Pierre
Download16m1109394.pdf (5.233Mb)
PUBLISHER_POLICY
Publisher Policy
Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
Terms of use
Metadata
Show full item recordAbstract
Quantifying the uncertainty of Lagrangian motion can be performed by solving a large number of ordinary differential equations with random velocities or, equivalently, a stochastic transport partial differential equation (PDE) for the ensemble of flow-maps. The dynamically orthogonal (DO) decomposition is applied as an efficient dynamical model order reduction to solve for such stochastic advection and Lagrangian transport. Its interpretation as the method that applies the truncated SVD instantaneously on the matrix discretization of the original stochastic PDE is used to obtain new numerical schemes. Fully linear, explicit central advection schemes stabilized with numerical filters are selected to ensure efficiency, accuracy, stability, and direct consistency between the original deterministic and stochastic DO advections and flow-maps. Various strategies are presented for selecting a time-stepping that accounts for the curvature of the fixed-rank manifold and the error related to closely singular coefficient matrices. Efficient schemes are developed to dynamically evolve the rank of the reduced solution and to ensure the orthogonality of the basis matrix while preserving its smooth evolution over time. Finally, the new schemes are applied to quantify the uncertain Lagrangian motions of a 2D double-gyre flow with random frequency and of a stochastic flow past a cylinder. Keywords: dynamically orthogonal decomposition, stochastic advection, singular value decomposition, uncertainty quantification, flow-map, Lagrangian coherent structures
Date issued
2018-01Department
Massachusetts Institute of Technology. Department of Mechanical EngineeringJournal
SIAM Review
Publisher
Society for Industrial and Applied Mathematics
Citation
Feppon, Florian, and Pierre F. J. Lermusiaux. “Dynamically Orthogonal Numerical Schemes for Efficient Stochastic Advection and Lagrangian Transport.” SIAM Review 60, no. 3 (January 2018): 595–625. © 2018 Society for Industrial and Applied Mathematics.
Version: Final published version
ISSN
0036-1445
1095-7200