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A dual consistent summation-by-parts formulation for the linearized incompressible Navier-Stokes equations posed on deforming domains
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
2016 (English)Report (Other academic)
Abstract [en]

In this article, well-posedness and dual consistency of the linearized incompressible Navier-Stokes equations posed on time-dependent spatial domains are studied. To simplify the derivation of the dual problem, the second order formulation is transformed to rst order form. Boundary conditions that simultaneously lead to well-posedness of the primal and dual problems are derived.

We construct fully discrete nite di erence schemes on summation-byparts form, in combination with the simultaneous approximation technique. We prove energy stability and discrete dual consistency. Moreover, we show how to construct the penalty operators such that the scheme automatically adjusts to the variations of the spatial domain, and as a result, stability and discrete dual consistency follow simultaneously.

The method is illustrated by considering a deforming time-dependent spatial domain in two dimensions. The numerical calculations are performed using high order operators in space and time. The results corroborate the stability of the scheme and the accuracy of the solution. We also show that linear functionals are superconverging. Additionally, we investigate the convergence of non-linear functionals and the divergence of the solution.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2016. , 26 p.
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2016:10
Keyword [en]
Incompressible Navier-Stokes equations, Deforming domain, Stability, Dual consistency, High order accuracy, Superconvergence
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-130584ISRN: LiTH-MAT-R--2016/10--SEOAI: oai:DiVA.org:liu-130584DiVA: diva2:953289
Available from: 2016-08-17 Created: 2016-08-17 Last updated: 2016-09-19Bibliographically approved
In thesis
1. Stable High Order Finite Difference Methods for Wave Propagation and Flow Problems on Deforming Domains
Open this publication in new window or tab >>Stable High Order Finite Difference Methods for Wave Propagation and Flow Problems on Deforming Domains
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

We construct stable, accurate and efficient numerical schemes for wave propagation and flow problems posed on spatial geometries that are moving, deforming, erroneously described or non-simply connected. The schemes are on Summation-by-Parts (SBP) form, combined with the Simultaneous Approximation Term (SAT) technique for imposing initial and boundary conditions. The main analytical tool is the energy method, by which well-posedness, stability and conservation are investigated. To handle the deforming domains, time-dependent coordinate transformations are used to map the problem to fixed geometries.

The discretization is performed in such a way that the Numerical Geometric Conservation Law (NGCL) is satisfied. Additionally, even though the schemes are constructed on fixed domains, time-dependent penalty formulations are necessary, due to the originally moving boundaries. We show how to satisfy the NGCL and present an automatic formulation for the penalty operators, such that the correct number of boundary conditions are imposed, when and where required.

For problems posed on erroneously described geometries, we investigate how the accuracy of the solution is affected. It is shown that the inaccurate geometry descriptions may lead to wrong wave speeds, a misplacement of the boundary condition, the wrong boundary operator or a mismatch of data. Next, the SBP-SAT technique is extended to time-dependent coupling procedures for deforming interfaces in hyperbolic problems. We prove conservation and stability and show how to formulate the penalty operators such that the coupling procedure is automatically adjusted to the variations of the interface location while the NGCL is preserved.

Moreover, dual consistent SBP-SAT schemes for the linearized incompressible Navier-Stokes equations posed on deforming domains are investigated. To simplify the derivations of the dual problem and incorporate the motions of the boundaries, the second order formulation is reduced to first order and the problem is transformed to a fixed domain. We prove energy stability and dual consistency. It is shown that the solution as well as the divergence of the solution converge with the design order of accuracy, and that functionals of the solution are superconverging.

Finally, initial boundary value problems posed on non-simply connected spatial domains are investigated. The new formulation increases the accuracy of the scheme by minimizing the use of multi-block couplings. In order to show stability, the spectrum of the semi-discrete SBP-SAT formulation is studied. We show that the eigenvalues have the correct sign, which implies stability, in combination with the SBP-SAT technique in time.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2016. 23 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1774
National Category
Computational Mathematics Mathematical Analysis Fluid Mechanics and Acoustics
Identifiers
urn:nbn:se:liu:diva-130928 (URN)10.3384/diss.diva-130928 (DOI)9789176857373 (Print) (ISBN)
Public defence
2016-09-30, Visionen, Hus B, Campus Valla, Linköping, 13:15 (English)
Opponent
Supervisors
Available from: 2016-08-31 Created: 2016-08-31 Last updated: 2016-09-01Bibliographically approved

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