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Uncertainty Quantification and Numerical Methods for Conservation Laws
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Conservation laws with uncertain initial and boundary conditions are approximated using a generalized polynomial chaos expansion approach where the solution is represented as a generalized Fourier series of stochastic basis functions, e.g. orthogonal polynomials or wavelets. The stochastic Galerkin method is used to project the governing partial differential equation onto the stochastic basis functions to obtain an extended deterministic system.

The stochastic Galerkin and collocation methods are used to solve an advection-diffusion equation with uncertain viscosity. We investigate well-posedness, monotonicity and stability for the stochastic Galerkin system. High-order summation-by-parts operators and weak imposition of boundary conditions are used to prove stability. We investigate the impact of the total spatial operator on the convergence to steady-state. 

Next we apply the stochastic Galerkin method to Burgers' equation with uncertain boundary conditions. An analysis of the truncated polynomial chaos system presents a qualitative description of the development of the solution over time. An analytical solution is derived and the true polynomial chaos coefficients are shown to be smooth, while the corresponding coefficients of the truncated stochastic Galerkin formulation are shown to be discontinuous. We discuss the problematic implications of the lack of known boundary data and possible ways of imposing stable and accurate boundary conditions.

We present a new fully intrusive method for the Euler equations subject to uncertainty based on a Roe variable transformation. The Roe formulation saves computational cost compared to the formulation based on expansion of conservative variables. Moreover, it is more robust and can handle cases of supersonic flow, for which the conservative variable formulation fails to produce a bounded solution. A multiwavelet basis that can handle  discontinuities in a robust way is used.

Finally, we investigate a two-phase flow problem. Based on regularity analysis of the generalized polynomial chaos coefficients, we present a hybrid method where solution regions of varying smoothness are coupled weakly through interfaces. In this way, we couple smooth solutions solved with high-order finite difference methods with non-smooth solutions solved for with shock-capturing methods.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2013. , 39 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1008
Keyword [en]
uncertainty quantification, polynomial chaos, stochastic Galerkin methods, conservation laws, hyperbolic problems, finite difference methods, finite volume methods
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
URN: urn:nbn:se:uu:diva-188348ISBN: 978-91-554-8569-6 (print)OAI: oai:DiVA.org:uu-188348DiVA: diva2:577676
Public defence
2013-02-08, Room 2446, Polacksbacken, Lägerhyddsvägen 2D, Uppsala, 10:15 (English)
Opponent
Supervisors
Available from: 2013-01-11 Created: 2012-12-16 Last updated: 2013-04-02Bibliographically approved
List of papers
1. Numerical analysis of the Burgers' equation in the presence of uncertainty
Open this publication in new window or tab >>Numerical analysis of the Burgers' equation in the presence of uncertainty
2009 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 228, 8394-8412 p.Article in journal (Refereed) Published
National Category
Computational Mathematics Computer Science
Identifiers
urn:nbn:se:uu:diva-108561 (URN)10.1016/j.jcp.2009.08.012 (DOI)000271342600011 ()
Available from: 2009-09-22 Created: 2009-09-22 Last updated: 2017-12-13Bibliographically approved
2. Boundary procedures for the time-dependent Burgers' equation under uncertainty
Open this publication in new window or tab >>Boundary procedures for the time-dependent Burgers' equation under uncertainty
2010 (English)In: Acta Mathematica Scientia, ISSN 0252-9602, E-ISSN 1003-3998, Vol. 30, 539-550 p.Article in journal (Refereed) Published
National Category
Computational Mathematics Computer Science
Identifiers
urn:nbn:se:uu:diva-123426 (URN)10.1016/S0252-9602(10)60061-6 (DOI)000276112800009 ()
Available from: 2010-04-02 Created: 2010-04-27 Last updated: 2017-12-12Bibliographically approved
3. On stability and monotonicity requirements of discretized stochastic conservation laws with random viscosity
Open this publication in new window or tab >>On stability and monotonicity requirements of discretized stochastic conservation laws with random viscosity
2012 (English)Report (Other academic)
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-028
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-182195 (URN)
Available from: 2012-09-30 Created: 2012-10-04 Last updated: 2013-01-11Bibliographically approved
4. A stochastic Galerkin method for the Euler equations with Roe variable transformation
Open this publication in new window or tab >>A stochastic Galerkin method for the Euler equations with Roe variable transformation
2012 (English)Report (Other academic)
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-033
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-184967 (URN)
Available from: 2012-11-15 Created: 2012-11-15 Last updated: 2013-01-11Bibliographically approved
5. An intrusive hybrid method for discontinuous two-phase flow under uncertainty
Open this publication in new window or tab >>An intrusive hybrid method for discontinuous two-phase flow under uncertainty
2012 (English)Report (Other academic)
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-035
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-188347 (URN)
Available from: 2012-12-16 Created: 2012-12-16 Last updated: 2013-01-11Bibliographically approved

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