Intrusive methods in uncertainty quantification and their connection to kinetic theory

Intrusive methods in uncertainty quantification and their connection to kinetic theory Uncertainty quantification for hyperbolic equations is a challenging task, since solutions exhibit discontinuities and sharp gradients. The commonly used stochastic-Galerkin (SG) Method uses polynomials to represent the solution, leading to oscillatory approximations due to Gibbs phenomenon. Additionally, the SG moment systems can loose hyperbolicity. The intrusive polynomial moment method (IPMM) yields a general framework for intrusive methods while ensuring hyperbolicity of the moment system and restricting oscillatory over- and undershoots to specified bounds. In this contribution, similarities as well as differences of the IPMM to minimal entropy closures used in transport theory will be discussed. We apply filters, which are well-known in kinetic theory to the SG Method to limit oscillatory overshoots. By investigating Burgers’ equation, we demonstrate that the use of filters improves the results of stochastic Galerkin. The IPMM method shows better approximation results than the filter, but comes at a higher computational cost. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png International Journal of Advances in Engineering Sciences and Applied Mathematics Springer Journals

Intrusive methods in uncertainty quantification and their connection to kinetic theory

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Indian Institute of Technology Madras
Subject
Engineering; Engineering, general; Mathematical and Computational Engineering
ISSN
0975-0770
eISSN
0975-5616
D.O.I.
10.1007/s12572-018-0211-3
Publisher site
See Article on Publisher Site

Abstract

Uncertainty quantification for hyperbolic equations is a challenging task, since solutions exhibit discontinuities and sharp gradients. The commonly used stochastic-Galerkin (SG) Method uses polynomials to represent the solution, leading to oscillatory approximations due to Gibbs phenomenon. Additionally, the SG moment systems can loose hyperbolicity. The intrusive polynomial moment method (IPMM) yields a general framework for intrusive methods while ensuring hyperbolicity of the moment system and restricting oscillatory over- and undershoots to specified bounds. In this contribution, similarities as well as differences of the IPMM to minimal entropy closures used in transport theory will be discussed. We apply filters, which are well-known in kinetic theory to the SG Method to limit oscillatory overshoots. By investigating Burgers’ equation, we demonstrate that the use of filters improves the results of stochastic Galerkin. The IPMM method shows better approximation results than the filter, but comes at a higher computational cost.

Journal

International Journal of Advances in Engineering Sciences and Applied MathematicsSpringer Journals

Published: Jun 1, 2018

References

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