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A stochastic Galerkin method for the Euler equations with Roe variable transformation
Institute for Computational and Mathematical Engineering and Department of Energy Resources Engineering, Stanford University, Stanford, CA, USA, Department of Information Technology, Uppsala University, Uppsala, Sweden.
Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA, USA.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.ORCID iD: 0000-0002-7972-6183
2014 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 257, no A, 481-500 p.Article in journal (Refereed) Published
Abstract [en]

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 chosen 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.

Place, publisher, year, edition, pages
Elsevier, 2014. Vol. 257, no A, 481-500 p.
Keyword [en]
Uncertainty quantification, Euler equations, Roe variable transformation, Stochastic Galerkin method, Multi-wavelets
National Category
Computational Mathematics
URN: urn:nbn:se:liu:diva-99007DOI: 10.1016/ 000327483200023OAI: diva2:656262
Available from: 2013-10-15 Created: 2013-10-15 Last updated: 2014-04-16Bibliographically approved

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