On stability and monotonicity requirements of finite difference approximations of stochastic conservation laws with random viscosity

Handle URI:
http://hdl.handle.net/10754/599044
Title:
On stability and monotonicity requirements of finite difference approximations of stochastic conservation laws with random viscosity
Authors:
Pettersson, Per; Doostan, Alireza; Nordström, Jan
Abstract:
The stochastic Galerkin and collocation methods are used to solve an advection-diffusion 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-diffusion equation onto the stochastic basis functions. High-order summation-by-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 steady-state. © 2013 Elsevier B.V.
Citation:
Pettersson P, Doostan A, Nordström J (2013) On stability and monotonicity requirements of finite difference approximations of stochastic conservation laws with random viscosity. Computer Methods in Applied Mechanics and Engineering 258: 134–151. Available: http://dx.doi.org/10.1016/j.cma.2013.02.009.
Publisher:
Elsevier BV
Journal:
Computer Methods in Applied Mechanics and Engineering
Issue Date:
May-2013
DOI:
10.1016/j.cma.2013.02.009
Type:
Article
ISSN:
0045-7825
Sponsors:
The first author gratefully acknowledges funding from King Abdullah University of Science and Technology (KAUST), Saudi Arabia. The second author gratefully acknowledges the support of the Department of Energy under grant DE-SC0006402.
Appears in Collections:
Publications Acknowledging KAUST Support

Full metadata record

DC FieldValue Language
dc.contributor.authorPettersson, Peren
dc.contributor.authorDoostan, Alirezaen
dc.contributor.authorNordström, Janen
dc.date.accessioned2016-02-25T13:51:47Zen
dc.date.available2016-02-25T13:51:47Zen
dc.date.issued2013-05en
dc.identifier.citationPettersson P, Doostan A, Nordström J (2013) On stability and monotonicity requirements of finite difference approximations of stochastic conservation laws with random viscosity. Computer Methods in Applied Mechanics and Engineering 258: 134–151. Available: http://dx.doi.org/10.1016/j.cma.2013.02.009.en
dc.identifier.issn0045-7825en
dc.identifier.doi10.1016/j.cma.2013.02.009en
dc.identifier.urihttp://hdl.handle.net/10754/599044en
dc.description.abstractThe stochastic Galerkin and collocation methods are used to solve an advection-diffusion 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-diffusion equation onto the stochastic basis functions. High-order summation-by-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 steady-state. © 2013 Elsevier B.V.en
dc.description.sponsorshipThe first author gratefully acknowledges funding from King Abdullah University of Science and Technology (KAUST), Saudi Arabia. The second author gratefully acknowledges the support of the Department of Energy under grant DE-SC0006402.en
dc.publisherElsevier BVen
dc.subjectMonotonicityen
dc.subjectPolynomial chaosen
dc.subjectStabilityen
dc.subjectStochastic collocationen
dc.subjectStochastic Galerkinen
dc.subjectSummation-by-parts operatorsen
dc.titleOn stability and monotonicity requirements of finite difference approximations of stochastic conservation laws with random viscosityen
dc.typeArticleen
dc.identifier.journalComputer Methods in Applied Mechanics and Engineeringen
dc.contributor.institutionStanford University, Palo Alto, United Statesen
dc.contributor.institutionUniversity of Colorado at Boulder, Boulder, United Statesen
dc.contributor.institutionLinkopings universitet, Linkoping, Swedenen
dc.contributor.institutionUppsala Universitet, Uppsala, Swedenen
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