Dynamically Orthogonal Numerical Schemes for Efficient Stochastic Advection and Lagrangian Transport

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) decomposit...

Full description

Saved in:
Bibliographic Details
Published in:SIAM review 2018-09, Vol.60 (3), p.595-625
Main Authors: Feppon, Florian, Lermusiaux, Pierre F. J.
Format: Article
Language:English
Subjects:
Analysis of PDEs
Differential Geometry
Dynamical Systems
Mathematics
Numerical Analysis
RESEARCH SPOTLIGHTS
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:0036-1445
1095-7200
DOI:10.1137/16M1109394