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Stability, dual consistency and conservation of summation-by-parts formulations for multiphysics problems
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis, we consider the numerical solution of initial boundary value problems (IBVPs). Boundary and interface conditions are derived such that the IBVP under consideration is well-posed. We also study the dual problem and the related dual boundary/interface conditions. Once the continuous problem is analyzed, we use finite difference operators with the Summation- By-Parts property (SBP) and a weak boundary/interface treatment using the Simultaneous-Approximation-Terms (SAT) technique to construct high-order accurate numerical schemes. We focus in particular on stability, conservation and dual consistency. The energy method is used as our main analysis tool for both the continuous and numerical problems.

The contributions of this thesis can be divided into two parts. The first part focuses on the coupling of different IBVPs. Interface conditions are derived such that the continuous problem satisfy an energy estimate and such that the discrete problem is stable. In the first paper, two hyperbolic systems of different size posed on two domains are considered. We derive the dual problem and dual interface conditions. It is also shown that a specific choice of penalty matrices leads to dual consistency. As an application, we study the coupling of the Euler and wave equations. In the fourth paper, we examine how to couple the compressible and incompressible Navier-Stokes equations. In order to obtain a sufficient number of interface conditions, the decoupled heat equation is added to the incompressible equations. The interface conditions include mass and momentum balance and two variants of heat transfer. The typical application in this case is the atmosphere-ocean coupling.

The second part of the thesis focuses on the relation between the primal and dual problem and the relation between dual consistency and conservation. In the second and third paper, we show that dual consistency and conservation are equivalent concepts for linear hyperbolic conservation laws. We also show that these concepts are equivalent for symmetric or symmetrizable parabolic problems in the fifth contribution. The relation between the primal and dual boundary conditions for linear hyperbolic systems of equations is investigated in the sixth and last paper. It is shown that for given well-posed primal/dual boundary conditions, the corresponding well-posed dual/primal boundary conditions can be obtained by a simple scaling operation. It is also shown how one can proceed directly from the well-posed weak primal problem to the well-posed weak dual problem.  

Abstract [sv]

Den här avhandlingen handlar om numeriska metoder för att lösa initial och randvärdes problem. Studien fokuserar på härledningen av rand/kopplingsvillkor som garanterar välställdhet. Det duala problemet och dess duala rand/kopplingsvillkor studeras också. Dessa problem diskretiseras genom att använda noggranna finita differensscheman på SBP-form (eng. summation-by-parts), kombinerat med en svag randbehandling benämnd SAT (eng. simultaneous approximation term). Vi fokuserar särskilt på stabilitet, konservation och dualkonsistens. Det främsta analysverktyget för både det kontinuerliga och diskreta problemet är energimetoden.

Den första delen av avhandlingen behandlar välställdhet och stabilitet för koppling av olika system av ekvationer. Kopplingsvillkoren är härledda så att det kontinuerliga problemet uppfyller en energiuppskattning och så att det diskreta problemet är stabilt. I den första artikeln görs analysen för koppling av två olika hyperboliska system på första ordningens form. Som tillämpning kopplar vi Euler och vågekvationerna. Koppling mellan kompressibla och inkompressibla Navier-Stokes ekvationer studeras i den fjärde artikeln. För att få rätt antal kopplingsvillkor lägger vi till värmeledningsekvationen till de inkompressibla ekvationerna. Kopplingsvillkoren innefattar massans och rörelsemängdens bevarande samt två varianter av värmeöverföring. Den typiska tillämpningen är koppling mellan atmosfär och hav.

Den andra delen av avhandlingen fokuserar på relationen mellan det primära och duala problemet och relationen mellan dualkonsistens och konservation. I den andra och tredje artikeln visar vi att dualkonsistens och konservation är ekvivalenta koncept för linjära hyperboliska konserveringslagar. I den femte artikeln, visas att dessa koncept är ekvivalenta även för paraboliska problem. Relationen mellan de primära och duala randvilkoren för linjära hyperboliska system av ekvationer i två dimensioner studeras i den sista artikeln. Vi visar att primära/duala välställda randvilkor ger duala/primära välställda randvilkor genom en enkel skalningsoperation. Det visas också att man kan gå direkt från det välställda svaga primära problemet till det välställda svaga duala problemet.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2019. , p. 27
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1998
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-159189DOI: 10.3384/diss.diva-159189ISBN: 9789176850312 (print)OAI: oai:DiVA.org:liu-159189DiVA, id: diva2:1339992
Public defence
2019-09-13, Ada Lovelace, B-huset, Campus Valla, Linköping, 13:15 (English)
Opponent
Supervisors
Available from: 2019-08-02 Created: 2019-08-01 Last updated: 2019-11-01Bibliographically approved
List of papers
1. Coupling Requirements for Multiphysics Problems Posed on Two Domains
Open this publication in new window or tab >>Coupling Requirements for Multiphysics Problems Posed on Two Domains
2017 (English)In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 55, no 6, p. 2885-2904Article in journal (Refereed) Published
Abstract [en]

We consider two hyperbolic systems in first order form of different size posed on two domains. Our ambition is to derive general conditions for when the two systems can and cannot be coupled. The adjoint equations are derived and well-posedness of the primal and dual problems is discussed. By applying the energy method, interface conditions for the primal and dual problems are derived such that the continuous problems are well posed. The equations are discretized using a high order finite difference method in summation-by-parts form and the interface conditions are imposed weakly in a stable way, using penalty formulations. It is shown that one specic choice of penalty matrices leads to a dual consistent scheme. By considering an example, it is shown that the correct physical coupling conditions are contained in the set of well posed coupling conditions. It is also shown that dual consistency leads to superconverging functionals and reduced stiffness.

Place, publisher, year, edition, pages
Society for Industrial and Applied Mathematics, 2017
Keywords
well posed problems, high order finite diffrences, stability, summation-by-parts, weak interface conditions, dual consistency, stiffness, superconvergence
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-143261 (URN)10.1137/16M1087710 (DOI)000418663500015 ()
Available from: 2017-11-28 Created: 2017-11-28 Last updated: 2019-08-01Bibliographically approved
2. On the relation between conservation and dual consistency for summation-by-parts schemes
Open this publication in new window or tab >>On the relation between conservation and dual consistency for summation-by-parts schemes
2017 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 344, p. 3p. 437-439Article in journal (Refereed) Published
Abstract [en]

n/a

Publisher
p. 3
Keywords
Initial boundary value problems Summation-by-parts Conservation, Dual consistent, Multi-block, Multi-element
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-137545 (URN)10.1016/j.jcp.2017.04.072 (DOI)000402481300021 ()
Note

Classified in the journal as "Short note"

Available from: 2017-05-21 Created: 2017-05-21 Last updated: 2019-08-01
3. Corrigendum to “On the relation between conservation and dual consistency for summation-by-parts schemes”[J. Comput. Phys. 344 (2017) 437–439]
Open this publication in new window or tab >>Corrigendum to “On the relation between conservation and dual consistency for summation-by-parts schemes”[J. Comput. Phys. 344 (2017) 437–439]
2018 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 360, p. 247-247Article in journal (Refereed) Published
Place, publisher, year, edition, pages
Academic Press, 2018
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-145718 (URN)10.1016/j.jcp.2018.02.046 (DOI)000428966300014 ()
Available from: 2018-03-19 Created: 2018-03-19 Last updated: 2019-08-01Bibliographically approved
4. An energy stable coupling procedure for the compressible and incompressible Navier-Stokes equations
Open this publication in new window or tab >>An energy stable coupling procedure for the compressible and incompressible Navier-Stokes equations
2019 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 396, p. 280-302Article in journal (Refereed) Published
Abstract [en]

The coupling of the compressible and incompressible Navier-Stokes equations is considered. Our ambition is to take a first step towards a provably well posed and stable coupling procedure. We study a simplified setting with a stationary planar interface and small disturbances from a steady background flow with zero velocity normal to the interface. The simplified setting motivates the use of the linearized equations, and we derive interface conditions such that the continuous problem satisfy an energy estimate. The interface conditions can be imposed both strongly and weakly. It is shown that the weak and strong interface imposition produce similar continuous energy estimates. We discretize the problem in time and space by employing finite difference operators that satisfy a summation-by-parts rule. The interface and initial conditions are imposed weakly using a penalty formulation. It is shown that the results obtained for the weak interface conditions in the continuous case, lead directly to stability of the fully discrete problem.

Place, publisher, year, edition, pages
Elsevier, 2019
Keywords
Compressible fluid, Incompressible fluid, Navier-Stokes equations, Energy estimate, Interface conditions, Stability
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-159116 (URN)10.1016/j.jcp.2019.07.022 (DOI)000481732600015 ()
Available from: 2019-07-29 Created: 2019-07-29 Last updated: 2019-09-09
5. On conservation and dual consistency for summation-by-parts based approximations of parabolic problems
Open this publication in new window or tab >>On conservation and dual consistency for summation-by-parts based approximations of parabolic problems
2019 (English)Report (Other academic)
Abstract [en]

We consider the coupling of parabolic problems discretized using difference operators on summation-by-parts (SBP) form with interface conditions imposed weakly. In, it was shown that conservation and dual consistency are equivalent concepts for linear conservation laws. Here, we show that these concepts are equivalent also for symmetric or symmetrizable parabolic problems, exemplified by the heat equation. We rewrite the heat equation as first order system as in the local discontinuous Galerkin method and show the equivalence of dual consistency and conservation for both the first and second order forms.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2019. p. 9
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2019:5
National Category
Computational Mathematics Mathematics
Identifiers
urn:nbn:se:liu:diva-158903 (URN)LiTH-MAT-R--2019/05--SE (ISRN)
Available from: 2019-07-17 Created: 2019-07-17 Last updated: 2019-08-12Bibliographically approved
6. The relation between primal and dual boundary conditions for hyperbolic systems of equations
Open this publication in new window or tab >>The relation between primal and dual boundary conditions for hyperbolic systems of equations
2020 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 401, article id 109032Article in journal (Refereed) Published
Abstract [en]

In this paper we study boundary conditions for linear hyperbolic systems of equations and the corresponding dual problem. In particular, we show that the primal and dual boundary conditions are related by a simple scaling relation. It is also shown that the weak dual problem can be derived directly from the weak primal problem. Based on the continuous analysis, we discretize and perform computations with a high-order finite difference scheme on summation- by-parts form with weak boundary conditions. It is shown that the results obtained in the continuous analysis lead directly to stability results for the primal and dual discrete problems. Numerical experiments corroborate the theoretical results.

Place, publisher, year, edition, pages
Elsevier, 2020
Keywords
Hyperbolic systems, Boundary conditions, Primal problem, Dual problem, Well-posedness, Dual consistency
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-161459 (URN)10.1016/j.jcp.2019.109032 (DOI)
Available from: 2019-11-01 Created: 2019-11-01 Last updated: 2019-12-05Bibliographically approved

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Ghasemi Zinatabadi, Fatemeh

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