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Numerical analysis of the Burgers' equation in the presence of uncertainty
Department of Mechanical Engineering, Stanford University, Stanford, USA; Uppsala universitet, Avdelningen för teknisk databehandling.
Department of Mechanical Engineering, Stanford University, Stanford, USA.
Uppsala universitet, Avdelningen för teknisk databehandling; Department of Aeronautics and Systems Integration, FOI, Stockholm, Sweden; School of Mechanical, Industrial and Aeronautical Engineering, University of the Witwatersrand, Johannesburg, South Africa.ORCID iD: 0000-0002-7972-6183
2009 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 228, 8394-8412 p.Article in journal (Refereed) Published
Abstract [en]

The Burgers’ equation with uncertain initial and boundary conditions is investigated usinga polynomial chaos (PC) expansion approach where the solution is represented as a truncatedseries of stochastic, orthogonal polynomials.The analysis of well-posedness for the system resulting after Galerkin projection is presentedand follows the pattern of the corresponding deterministic Burgers equation. Thenumerical discretization is based on spatial derivative operators satisfying the summationby parts property and weak boundary conditions to ensure stability. Similarly to the deterministiccase, the explicit time step for the hyperbolic stochastic problem is proportional tothe inverse of the largest eigenvalue of the system matrix. The time step naturallydecreases compared to the deterministic case since the spectral radius of the continuousproblem grows with the number of polynomial chaos coefficients. An estimate of theeigenvalues is provided.A characteristic analysis of the truncated PC system is presented and gives a qualitativedescription of the development of the system over time for different initial and boundaryconditions. It is shown that a precise statistical characterization of the input uncertainty isrequired and partial information, e.g. the expected values and the variance, are not sufficientto obtain a solution. An analytical solution is derived and the coefficients of the infinitePC expansion are shown to be smooth, while the corresponding coefficients of thetruncated expansion are discontinuous.

Place, publisher, year, edition, pages
2009. Vol. 228, 8394-8412 p.
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Computational Mathematics Computer Science
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URN: urn:nbn:se:liu:diva-68604DOI: 10.1016/j.jcp.2009.08.012ISI: 000271342600011OAI: oai:DiVA.org:liu-68604DiVA: diva2:418726
Available from: 2009-09-22 Created: 2011-05-24 Last updated: 2017-12-11

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Pettersson, PerNordström, Jan

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