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High order summation-by-parts based approximations for discontinuous and nonlinear problems
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Numerical approximations using high order finite differences on summation-byparts (SBP) form are investigated for discontinuous and fully nonlinear systems of partial differential equations. Stability and conservation properties of the approximations are obtained through a weak imposition of interface and boundary conditions with the simultaneous-approximation-term (SAT) technique. The SBP-SAT approximations replicate the continuous integration by parts rule. From this property, well-posedness and integral properties of the continuous problem are mimicked, and energy estimates leading to stability are obtained.

The first part of the thesis focuses on the simulations of discontinuous linear advection problems. An artificial interface is introduced, separating parts of the spatial domain characterized by different wave speeds. A set of flexible stability conditions at the interface are derived, which can be adapted to yield conservative or non-conservative approximations. This model can be interpreted as a simplified version of nonlinear problems involving jumps at shocks, or as a prototypical of wave propagation through different materials.

In the second part of the thesis, the vorticity/stream function formulation of the nonlinear momentum equation for an incompressible inviscid fluid is considered. SBP operators are used to derive a new Arakawa-like Jacobian with mimetic properties by combining different consistent approximations of the convection terms. Energy and enstrophy conservation is obtained for periodic problems using schemes with arbitrarily high order of accuracy. These properties are crucial for long-term numerical calculations in climate and weather forecasts or ocean circulation predictions.

The third and final contribution of the thesis is dedicated to the incompressible Navier-Stokes problem. First, different completely general formulations of energy bounding boundary conditions are derived for the nonlinear equations. The boundary conditions can be used at both far field and solid wall boundaries. The discretisation in time and space with weakly imposed initial and boundary conditions using the SBP-SAT framework is proved to be stable and the divergence free condition is approximated with the design order of the scheme. Next, the same formulations are considered in a linearised setting, whereupon the spectra associated with the initial boundary value problem and its SBP-SAT discretisation are derived using the Laplace-Fourier technique. The influence of different boundary conditions on the spectrum and in particular the convergence to steady state is studied.

Abstract [sv]

Numeriska approximationer av ekvationer som styr fysikaliska lagar är avgörande i många tillämpningar. Förutom en matematisk modell som kan fånga huvuddragen i ett verkligt problem är det nödvändigt att kunna utföra tillförlitliga simuleringar.

Denna avhandling behandlar numeriska approximationer som med hög noggrannhet bevarar både rent matematiska aspekter av ekvationerna så väl som viktiga egenskaper hos modellen. Dessutom ges särskild uppmärksamhet åt modeller med diskontinuiteter och icke-linjära beteenden.

Den första delen av avhandlingen handlar om diskontinuerliga problem. Det fysiska rummet kan ha olika egenskaper i olika regioner, något som kan resultera i instabila lösningar. Tillvägagångssättet består av att införa artificiella gränssnitt som skiljer dessa regioneråt. På detta sätt kan varje region behandlas separat, men på liknande sätt. Exempel på naturliga tillämningsområden är vågutbredning genom olika material och jordbävningssimuleringar.

I den andra delen av avhandlingen visar vi att om den numeriska approximationen imiterar partiell integration, då följer också de väsentliga egenskaperna hos modellen på ett naturligt sätt. Att fysikaliska egenskaper bevaras är nödvändigt för att bibehålla stabilitet under långa simuleringstider för bland annat geofysiska problem.

Den sista delen av avhandlingen är ägnasåt en av de mest använda modellerna inom strömningsmekanik, nämligen Navier-Stokes ekvationer. Studien fokuserar på härledningen av randvillkor som garanterar att lösningen inte växer på ett oförutsett och okontrollerat vis. Slutligen visas att de härledda randvillkoren på ett korrekt och noggrant sätt återskapar den dissipativa mekanism som ger upphov till jämviktstillstånd.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2017. , 38 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1880
National Category
Computational Mathematics Mathematical Analysis Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-140497DOI: 10.3384/diss.diva-140497ISBN: 9789176854525 (print)OAI: oai:DiVA.org:liu-140497DiVA: diva2:1138469
Public defence
2017-10-06, Ada Lovelace (former Visionen), B building, Campus Valla, Linköping, 13:15 (English)
Opponent
Supervisors
Available from: 2017-09-05 Created: 2017-09-05 Last updated: 2017-09-14Bibliographically approved
List of papers
1. Well-posedness, stability and conservation for a discontinuous interface problem
Open this publication in new window or tab >>Well-posedness, stability and conservation for a discontinuous interface problem
2016 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 56, no 2, 681-704 p.Article in journal (Refereed) Published
Abstract [en]

The advection equation is studied in a completely general two domain setting with different wave-speeds and a time-independent jump-condition at the interface separating the domains. Well-posedness and conservation criteria are derived for the initial-boundary-value problem. The equations are semi-discretized using a finite difference method on Summation-By-Part (SBP) form. The relation between the stability and conservation properties of the approximation are studied when the boundary and interface conditions are weakly imposed by the Simultaneous-Approximation-Term (SAT) procedure. Numerical simulations corroborate the theoretical findings.

Place, publisher, year, edition, pages
Springer, 2016
Keyword
Interface – Discontinuous coefficients problems – Initial boundary value problems – Well-posedness – Conservation – Stability – Interface conditions – High order accuracy – Summation-by-parts operators
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-121468 (URN)10.1007/s10543-015-0576-7 (DOI)000376580200016 ()
Available from: 2015-09-21 Created: 2015-09-21 Last updated: 2017-09-05Bibliographically approved
2. A new high order energy and enstrophy conserving Arakawa-like Jacobian differential operator
Open this publication in new window or tab >>A new high order energy and enstrophy conserving Arakawa-like Jacobian differential operator
2015 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 301, 167-177 p.Article in journal (Refereed) Published
Abstract [en]

A new high order Arakawa-like method for the incompressible vorticity equation in two-dimensions has been developed. Mimetic properties such as skew-symmetry, energy and enstrophy conservations for the semi-discretization are proved for periodic problems using arbitrary high order summation-by-parts operators. Numerical simulations corroborate the theoretical findings.

Place, publisher, year, edition, pages
Elsevier, 2015
Keyword
Non-linear problems; Summation-by-parts operators; Jacobian; Mimetic schemes; High-order schemes; Stability; Finite difference
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-121469 (URN)10.1016/j.jcp.2015.08.028 (DOI)000362379300010 ()
Available from: 2015-09-21 Created: 2015-09-21 Last updated: 2017-09-05
3. Energy Stable Boundary Conditions for the Nonlinear Incompressible Navier-Stokes Equations
Open this publication in new window or tab >>Energy Stable Boundary Conditions for the Nonlinear Incompressible Navier-Stokes Equations
2017 (English)Report (Other academic)
Abstract [en]

The nonlinear incompressible Navier-Stokes equations with boundary conditions at far fields and solid walls is considered. Two different formulations of boundary conditions are derived using the energy method. Both formulations are implemented in both strong and weak form and lead to an estimate of the velocity field. Equipped with energy bounding boundary conditions, the problem is approximated by using difference operators on summation-by-parts form and weak boundary and initial conditions. By mimicking the continuous analysis, the resulting semi-discrete as well as fully discrete scheme are shown to be provably stable, divergence free and high-order accurate.

Place, publisher, year, edition, pages
Linköping University Electronic Press, 2017. 31 p.
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2017:9
Keyword
Navier-Stokes equations, incompressible, boundary conditions, energy estimate, stability, summation-by-parts, high-order accuracy, divergence free
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-139730 (URN)LiTH-MAT-R--2017/09--SE (ISRN)
Available from: 2017-08-14 Created: 2017-08-14 Last updated: 2017-09-05Bibliographically approved
4. Spectral analysis of the incompressible Navier-Stokes equations with different boundary conditions
Open this publication in new window or tab >>Spectral analysis of the incompressible Navier-Stokes equations with different boundary conditions
2017 (English)Report (Other academic)
Abstract [en]

The influence of boundary conditions on the spectrum of the incompressible Navier-Stokes equations is studied. The spectra associated to different types of boundary conditions are derived using the Fourier-Laplace technique. In particular, the effect of various combinations of generalized in- and outgoing variables on the convergence to the steady state is investigated. The boundary conditions are analysed in both the continuous and semi-discrete problems. In the latter, high-order schemes in summation-by-parts form with weakly imposed boundary conditions are used to approximate the equations. Numerical calculations are performed and show that the discrete behaviour agrees with the theoretical analysis.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2017. 34 p.
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2017:10
Keyword
Incompressible flows, Navier-Stokes, Fourier-Laplace technique, summation-by-parts, weak boundary conditions
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-139729 (URN)LiTH-MAT-R--2017/10—SE (ISRN)
Available from: 2017-08-14 Created: 2017-08-14 Last updated: 2017-09-05Bibliographically approved

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