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On Stability and Monotonicity Requirements of Finite Difference Approximations of Stochastic Conservation Laws with Random Viscosity
Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA 94305, USA, Department of Information Technology, Uppsala University, P.O. Box 337, SE-75105 Uppsala, Sweden.
Aerospace Engineering Science Department, University of Colorado, Boulder, CO 80309, USA.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.ORCID iD: 0000-0002-7972-6183
2013 (English)Report (Other academic)
Abstract [en]

The stochastic Galerkin and collocation methods are used to solve an advection-diusion equation with uncertain and spatially varying viscosity. We investigate well-posedness, monotonicity and stability for the extended system resulting from the Galerkin projection of the advection-diusion equation onto the stochastic basis functions. High-order summationby- parts operators and weak imposition of boundary conditions are used to prove stability of the semi-discrete system.

It is essential that the eigenvalues of the resulting viscosity matrix of the stochastic Galerkin system are positive and we investigate conditions for this to hold. When the viscosity matrix is diagonalizable, stochastic Galerkin and stochastic collocation are similar in terms of computational cost, and for some cases the accuracy is higher for stochastic Galerkin provided that monotonicity requirements are met. We also investigate the total spatial operator of the semi-discretized system and its impact on the convergence to steadystate

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2013. , 37 p.
LiTH-MAT-R, ISSN 0348-2960 ; 2013:3
Keyword [en]
Polynomial chaos, Stochastic Galerkin, Stochastic collocation, Stability, Monotonicity, Summation-by-parts operators
National Category
Computational Mathematics
URN: urn:nbn:se:liu:diva-90995ISRN: LiTH-MAT-R--2013/03--SEOAI: diva2:615562
Available from: 2013-04-11 Created: 2013-04-11 Last updated: 2013-08-30Bibliographically approved

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