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Uncertainty Quantification in Computational Fluid Dynamics pp 1–57Cite as

Non-intrusive Uncertainty Propagation with Error Bounds for Conservation Laws Containing Discontinuities

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Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 92))

Abstract

The propagation of statistical model parameter uncertainty in the numerical approximation of nonlinear conservation laws is considered. Of particular interest are nonlinear conservation laws containing uncertain parameters resulting in stochastic solutions with discontinuities in both physical and random variable dimensions. Using a finite number of deterministic numerical realizations, our objective is the accurate estimation of output uncertainty statistics (e.g. expectation and variance) for quantities of interest such as functionals, graphs, and fields. Given the error in numerical realizations, error bounds for output statistics are derived that may be numerically estimated and included in the calculation of output statistics. Unfortunately, the calculation of output statistics using classical techniques such as polynomial chaos, stochastic collocation, and sparse grid quadrature can be severely compromised by the presence of discontinuities in random variable dimensions. An alternative technique utilizing localized piecewise approximation combined with localized subscale recovery is shown to significantly improve the quality of calculated statistics when discontinuities are present. The success of this localized technique motivates the development of the HYbrid Global and Adaptive Polynomial (HYGAP) method described in Sect. 4.4. HYGAP employs a high accuracy global approximation when the solution data varies smoothly in a random variable dimension and local adaptive polynomial approximation with local postprocessing when the solution is non-smooth. To illustrate strengths and weaknesses of classical and newly proposed uncertainty propagation methods, a number of computational fluid dynamics (CFD) model problems containing various sources of parameter uncertainty are calculated including 1-D Burgers’ equation, subsonic and transonic flow over 2-D single-element and multi-element airfoils, transonic Navier-Stokes flow over a 3-D ONERA M6 wing, and supersonic Navier-Stokes flow over a greatly simplified Saturn-V rocket.

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Notes

  1. 1.

    This does require the numerical solution of a scalar implicit function relation for each characteristic that is easily solved to any desired accuracy.

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Acknowledgements

The author acknowledges the support of the NASA Fundamental Aeronautics Program for supporting this work. Computing resources have been provided by the NASA Ames Advanced Supercomputing Center.

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Authors and Affiliations

  1. NASA Ames Research Center, Moffett Field, CA, 94035, USA

    Timothy Barth

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  1. Timothy Barth
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Correspondence to Timothy Barth .

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Editors and Affiliations

  1. Faculty of Aerospace Engineering, TU Delft, Delft, The Netherlands

    Hester Bijl

  2. d' Alembert Institute-CNRS, Université Pierre et Marie Curie - Paris VI, Paris, France

    Didier Lucor

  3. Seminar für Angewandte Mathematik, ETH Zürich, Zürich, Switzerland

    Siddhartha Mishra

  4. ETH Zürich Seminar für Angewandte Mathematik, Zürich, Switzerland

    Christoph Schwab

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Barth, T. (2013). Non-intrusive Uncertainty Propagation with Error Bounds for Conservation Laws Containing Discontinuities. In: Bijl, H., Lucor, D., Mishra, S., Schwab, C. (eds) Uncertainty Quantification in Computational Fluid Dynamics. Lecture Notes in Computational Science and Engineering, vol 92. Springer, Cham. https://doi.org/10.1007/978-3-319-00885-1_1

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