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Uncertainty Quantification and Numerical Methods for Conservation Laws
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Avdelningen för beräkningsvetenskap. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Numerisk analys.
2013 (engelsk)Doktoravhandling, med artikler (Annet vitenskapelig)
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.

sted, utgiver, år, opplag, sider
Uppsala: Acta Universitatis Upsaliensis, 2013. , s. 39
Serie
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1008
Emneord [en]
uncertainty quantification, polynomial chaos, stochastic Galerkin methods, conservation laws, hyperbolic problems, finite difference methods, finite volume methods
HSV kategori
Forskningsprogram
Beräkningsvetenskap med inriktning mot numerisk analys
Identifikatorer
URN: urn:nbn:se:uu:diva-188348ISBN: 978-91-554-8569-6 (tryckt)OAI: oai:DiVA.org:uu-188348DiVA, id: diva2:577676
Disputas
2013-02-08, Room 2446, Polacksbacken, Lägerhyddsvägen 2D, Uppsala, 10:15 (engelsk)
Opponent
Veileder
Tilgjengelig fra: 2013-01-11 Laget: 2012-12-16 Sist oppdatert: 2013-04-02bibliografisk kontrollert
Delarbeid
1. Numerical analysis of the Burgers' equation in the presence of uncertainty
Åpne denne publikasjonen i ny fane eller vindu >>Numerical analysis of the Burgers' equation in the presence of uncertainty
2009 (engelsk)Inngår i: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 228, s. 8394-8412Artikkel i tidsskrift (Fagfellevurdert) Published
HSV kategori
Identifikatorer
urn:nbn:se:uu:diva-108561 (URN)10.1016/j.jcp.2009.08.012 (DOI)000271342600011 ()
Tilgjengelig fra: 2009-09-22 Laget: 2009-09-22 Sist oppdatert: 2018-01-13bibliografisk kontrollert
2. Boundary procedures for the time-dependent Burgers' equation under uncertainty
Åpne denne publikasjonen i ny fane eller vindu >>Boundary procedures for the time-dependent Burgers' equation under uncertainty
2010 (engelsk)Inngår i: Acta Mathematica Scientia, ISSN 0252-9602, E-ISSN 1003-3998, Vol. 30, s. 539-550Artikkel i tidsskrift (Fagfellevurdert) Published
HSV kategori
Identifikatorer
urn:nbn:se:uu:diva-123426 (URN)10.1016/S0252-9602(10)60061-6 (DOI)000276112800009 ()
Tilgjengelig fra: 2010-04-02 Laget: 2010-04-27 Sist oppdatert: 2018-01-12bibliografisk kontrollert
3. On stability and monotonicity requirements of discretized stochastic conservation laws with random viscosity
Åpne denne publikasjonen i ny fane eller vindu >>On stability and monotonicity requirements of discretized stochastic conservation laws with random viscosity
2012 (engelsk)Rapport (Annet vitenskapelig)
Serie
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-028
HSV kategori
Identifikatorer
urn:nbn:se:uu:diva-182195 (URN)
Tilgjengelig fra: 2012-09-30 Laget: 2012-10-04 Sist oppdatert: 2013-01-11bibliografisk kontrollert
4. A stochastic Galerkin method for the Euler equations with Roe variable transformation
Åpne denne publikasjonen i ny fane eller vindu >>A stochastic Galerkin method for the Euler equations with Roe variable transformation
2012 (engelsk)Rapport (Annet vitenskapelig)
Serie
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-033
HSV kategori
Identifikatorer
urn:nbn:se:uu:diva-184967 (URN)
Tilgjengelig fra: 2012-11-15 Laget: 2012-11-15 Sist oppdatert: 2013-01-11bibliografisk kontrollert
5. An intrusive hybrid method for discontinuous two-phase flow under uncertainty
Åpne denne publikasjonen i ny fane eller vindu >>An intrusive hybrid method for discontinuous two-phase flow under uncertainty
2012 (engelsk)Rapport (Annet vitenskapelig)
Serie
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-035
HSV kategori
Identifikatorer
urn:nbn:se:uu:diva-188347 (URN)
Tilgjengelig fra: 2012-12-16 Laget: 2012-12-16 Sist oppdatert: 2013-01-11bibliografisk kontrollert

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