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Title: A stochastic Galerkin method for the Euler equations with Roe variable transformation

The Euler equations subject to uncertainty in the initial and boundary conditions are investigated via the stochastic Galerkin approach. We present a new fully intrusive method based on a variable transformation of the continuous equations. Roe variables are employed to get quadratic dependence in the flux function and a well-defined Roe average matrix that can be determined without matrix inversion. In previous formulations based on generalized polynomial chaos expansion of the physical variables, the need to introduce stochastic expansions of inverse quantities, or square roots of stochastic quantities of interest, adds to the number of possible different ways to approximate the original stochastic problem. We present a method where the square roots occur in the choice of variables, resulting in an unambiguous problem formulation. The Roe formulation saves computational cost compared to the formulation based on expansion of conservative variables. Moreover, the Roe formulation is more robust and can handle cases of supersonic flow, for which the conservative variable formulation fails to produce a bounded solution. For certain stochastic basis functions, the proposed method can be made more effective and well-conditioned. This leads to increased robustness for both choices of variables. We use a multi-wavelet basis that can be chosenmore » to include a large number of resolution levels to handle more extreme cases (e.g. strong discontinuities) in a robust way. For smooth cases, the order of the polynomial representation can be increased for increased accuracy.« less
Authors:
 [1] ;  [2] ;  [3] ;  [1] ;  [4]
  1. Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA 94305 (United States)
  2. (United States)
  3. (Sweden)
  4. Department of Mathematics, Computational Mathematics, Linköping University, SE-58183 Linköping (Sweden)
Publication Date:
OSTI Identifier:
22230846
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Computational Physics; Journal Volume: 257; Journal Issue: Part A; 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; ACCURACY; APPROXIMATIONS; BOUNDARY CONDITIONS; CHAOS THEORY; COMPARATIVE EVALUATIONS; EQUATIONS; MATHEMATICAL SOLUTIONS; POLYNOMIALS; STOCHASTIC PROCESSES; SUPERSONIC FLOW; TRANSFORMATIONS