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Title: A convergence study for SPDEs using combined Polynomial Chaos and Dynamically-Orthogonal schemes

We study the convergence properties of the recently developed Dynamically Orthogonal (DO) field equations [1] in comparison with the Polynomial Chaos (PC) method. To this end, we consider a series of one-dimensional prototype SPDEs, whose solution can be expressed analytically, and which are associated with both linear (advection equation) and nonlinear (Burgers equation) problems with excitations that lead to unimodal and strongly bi-modal distributions. We also propose a hybrid approach to tackle the singular limit of the DO equations for the case of deterministic initial conditions. The results reveal that the DO method converges exponentially fast with respect to the number of modes (for the problems considered) giving same levels of computational accuracy comparable with the PC method but (in many cases) with substantially smaller computational cost compared to stochastic collocation, especially when the involved parametric space is high-dimensional.
Authors:
 [1] ;  [2] ;  [1]
  1. Division of Applied Mathematics, Brown University, Providence, RI 02912 (United States)
  2. Courant Institute of Mathematical Sciences, New York University, NY 10012 (United States)
Publication Date:
OSTI Identifier:
22233608
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Computational Physics; Journal Volume: 245; Other Information: Copyright (c) 2013 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICAL METHODS AND COMPUTING; ADVECTION; CHAOS THEORY; COMPARATIVE EVALUATIONS; CONVERGENCE; EXCITATION; FIELD EQUATIONS; MATHEMATICAL SOLUTIONS; NONLINEAR PROBLEMS; ONE-DIMENSIONAL CALCULATIONS; PARTIAL DIFFERENTIAL EQUATIONS; POLYNOMIALS; STOCHASTIC PROCESSES