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Robust Boundary Conditions for Stochastic Incompletely Parabolic Systems of Equations
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0002-7972-6183
2016 (English)Report (Other academic)
Abstract [en]

We study an incompletely parabolic system in three space dimensions with stochastic boundary and initial data. We show how the variance of the solution can be manipulated by the boundary conditions, while keeping the mean value of the solution unaffected. Estimates of the variance of the solution is presented both analytically and numerically. We exemplify the technique by applying it to an incompletely parabolic model problem, as well as the one-dimensional compressible Navier-Stokes equations.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2016. , p. 39
Series
LiTH-MAT-R, ISSN 0348-2960 ; 19
Keyword [en]
uncertainty quantifcation, incompletely parabolic system, initial boundary value problems, stochastic data, variance reduction, robust design
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-133365ISRN: LiTH-MAT-R--2016/19--SEOAI: oai:DiVA.org:liu-133365DiVA, id: diva2:1059403
Available from: 2016-12-22 Created: 2016-12-22 Last updated: 2018-04-09Bibliographically approved
In thesis
1. Uncertainty quantification for wave propagation and flow problems with random data
Open this publication in new window or tab >>Uncertainty quantification for wave propagation and flow problems with random data
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we study partial differential equations with random inputs. The effects that different boundary conditions with random data and uncertain geometries have on the solution are analyzed. Further, comparisons and couplings between different uncertainty quantification methods are performed. The numerical simulations are based on provably strongly stable finite difference formulations based on summation-by-parts operators and a weak implementation of boundary and interface conditions.

The first part of this thesis treats the construction of variance reducing boundary conditions. It is shown how the variance of the solution can be manipulated by the choice of boundary conditions, and a close relation between the variance of the solution and the energy estimate is established. The technique is studied on both a purely hyperbolic system as well as an incompletely parabolic system of equations. The applications considered are the Euler, Maxwell's, and Navier--Stokes equations.

The second part focuses on the effect of uncertain geometry on the solution. We consider a two-dimensional advection-diffusion equation with a stochastically varying boundary. We transform the problem to a fixed domain where comparisons can be made. Numerical results are performed on a problem in heat transfer, where the frequency and amplitude of the prescribed uncertainty are varied.

The final part of the thesis is devoted to the comparison and coupling of different uncertainty quantification methods. An efficiency analysis is performed using the intrusive polynomial chaos expansion with stochastic Galerkin projection, and nonintrusive numerical integration. The techniques are compared using the non-linear viscous Burgers' equation. A provably stable coupling procedure for the two methods is also constructed. The general coupling procedure is exemplified using a hyperbolic system of equations.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2018. p. 26
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1921
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-146966 (URN)10.3384/diss.diva-146966 (DOI)9789176853399 (ISBN)
Public defence
2018-05-04, Ada Lovelace, B-huset, Campus Valla, Linköping, 13:15 (English)
Opponent
Supervisors
Available from: 2018-04-09 Created: 2018-04-09 Last updated: 2018-04-09Bibliographically approved

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