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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, Faculty of Science & Engineering.ORCID iD: 0000-0002-7972-6183
Aerospace Engineering Sciences, University of Colorado Boulder, CO 80309, USA..
2016 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 306, 92-116 p.Article in journal (Refereed) Published
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 sufficient to capture the stochastic solution for the problem considered.

We derive boundary conditions for the continuous form of the stochastic Galerkin formulation of the velocity and pressure equations. The resulting problem formulation leads to an energy estimate for the divergence. With suitable boundary data on the pressure and velocity, the energy estimate implies zero divergence of the velocity field.

Based on the analysis of the continuous equations, we present a semi-discretized 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 in the laminar flow regime 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
Elsevier, 2016. Vol. 306, 92-116 p.
Keyword [en]
Uncertainty quantication, Incompressible Navier-Stokes equations, Summation-by-parts operators, Stochastic Galerkin method, Boundary conditions
National Category
Computational Mathematics
URN: urn:nbn:se:liu:diva-122818DOI: 10.1016/ 000366157000006OAI: diva2:873934

Funding agencies: SUPRI-B at Stanford University; U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research [DE-SC0006402]; Uni Research, Norway

Available from: 2015-11-25 Created: 2015-11-25 Last updated: 2016-07-14

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