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A Well-posed and Stable Stochastic Galerkin Formulation of the Incompressible Navier-Stokes Equations with Random Data
Uni Research, N-5007 Bergen, Norway.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.ORCID iD: 0000-0002-7972-6183
Aerospace Engineering Sciences, University of Colorado Boulder, CO 80309, USA.
2015 (English)Report (Other academic)
Abstract [en]

We present a well-posed stochastic Galerkin formulation of the incompressible Navier-Stokes equations with uncertainty in model parameters or the initial and boundary conditions. The stochastic Galerkin method involves representation of the solution through generalized polynomial chaos expansion and projection of the governing equations onto stochastic basis functions, resulting in an extended system of equations. A relatively low-order generalized polynomial chaos expansion is sucient to capture the stochastic solution.

We derive boundary conditions for an energy estimate that leads to zero divergence of the velocity field. In other words, the incompressibility condition is not imposed directly in the problem formulation but is instead a consequence of the combination of the partial differential equations and the boundary conditions.

Based on the analysis of the continuous equations, we present a semidiscretized system where the spatial derivatives are approximated using finite difference operators with a summation-by-parts property. With a suitable choice of dissipative boundary conditions imposed weakly through penalty terms, the semi-discrete scheme is shown to be stable. Numerical experiments corroborate the theoretical results and we obtain high-order accurate results for the solution variables and the velocity divergence converges to zero as the mesh is refined.

Place, publisher, year, edition, pages
Linköping University Electronic Press, 2015. , 46 p.
LiTH-MAT-R, ISSN 0348-2960 ; 2015:06
Keyword [en]
Uncertainty quantication, Incompressible Navier-Stokes equations, Summation-by-parts operators, Stochastic Galerkin method, Boundary conditions
National Category
Computational Mathematics Mathematics
URN: urn:nbn:se:liu:diva-115604ISRN: LiTH-MAT-R--2015/06--SEOAI: diva2:795934
Available from: 2015-03-17 Created: 2015-03-17 Last updated: 2015-03-17

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