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Stable High Order Finite Difference Methods for Wave Propagation and Flow Problems on Deforming Domains
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
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: urn:nbn:se:liu:diva-130928DOI: 10.3384/diss.diva-130928ISBN: 9789176857373 (Print)OAI: oai:DiVA.org:liu-130928DiVA: diva2:956877
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
List of papers
1. Fully discrete energy stable high order finite difference methods for hyperbolic problems in deforming domains
Open this publication in new window or tab >>Fully discrete energy stable high order finite difference methods for hyperbolic problems in deforming domains
2015 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 291, 82-98 p.Article in journal (Refereed) Published
Abstract [en]

A time-dependent coordinate transformation of a constant coefficient hyperbolic system of equations which results in a variable coefficient system of equations is considered. By applying the energy method, well-posed boundary conditions for the continuous problem are derived. Summation-by-Parts (SBP) operators for the space and time discretization, together with a weak imposition of boundary and initial conditions using Simultaneously Approximation Terms (SATs) lead to a provable fully-discrete energy-stable conservative finite difference scheme. We show how to construct a time-dependent SAT formulation that automatically imposes boundary conditions, when and where they are required. We also prove that a uniform flow field is preserved, i.e. the Numerical Geometric Conservation Law (NGCL) holds automatically by using SBP-SAT in time and space. The developed technique is illustrated by considering an application using the linearized Euler equations: the sound generated by moving boundaries. Numerical calculations corroborate the stability and accuracy of the new fully discrete approximations.

Place, publisher, year, edition, pages
Elsevier, 2015
Keyword
Deforming domain; Initial boundary value problems; High order accuracy; Well-posed boundary conditions; Summation-by-parts operators; Stability; Convergence; Conservation; Numerical geometric conservation law; Euler equation; Sound propagation
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-117360 (URN)10.1016/j.jcp.2015.02.027 (DOI)000352230500006 ()
Available from: 2015-04-24 Created: 2015-04-24 Last updated: 2016-08-31
2. Hyperbolic systems of equations posed on erroneous curved domains
Open this publication in new window or tab >>Hyperbolic systems of equations posed on erroneous curved domains
2016 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 308, 438-442 p.Article in journal (Refereed) Published
Abstract [en]

The effect of an inaccurate geometry description on the solution accuracy of a hyperbolic problem is discussed. The inaccurate geometry can for example come from an imperfect CAD system, a faulty mesh generator, bad measurements or simply a misconception.

We show that inaccurate geometry descriptions might lead to the wrong wave speeds, a misplacement of the boundary conditions, to the wrong boundary operator and a mismatch of boundary data.

The errors caused by an inaccurate geometry description may affect the solution more than the accuracy of the specific discretization techniques used. In extreme cases, the order of accuracy goes to zero. Numerical experiments corroborate the theoretical results.

Place, publisher, year, edition, pages
Elsevier, 2016
Keyword
Hyperbolic systems; Erroneous curved domains; Inaccurate data; Convergence rate
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-123918 (URN)10.1016/j.jcp.2015.12.048 (DOI)000369086700021 ()
Available from: 2016-01-13 Created: 2016-01-13 Last updated: 2016-08-31Bibliographically approved
3. A fully discrete, stable and conservative summation-by-parts formulation for deforming interfaces
Open this publication in new window or tab >>A fully discrete, stable and conservative summation-by-parts formulation for deforming interfaces
2016 (English)Report (Other academic)
Abstract [en]

We introduce an interface/coupling procedure for hyperbolic problems posedon time-dependent curved multi-domains. First, we transform the problem from Cartesian to boundary-conforming curvilinear coordinates and apply the energy method to derive well-posed and conservative interface conditions.

Next, we discretize the problem in space and time by employing finite difference operators that satisfy a summation-by-parts rule. The interface condition is imposed weakly using a penalty formulation. We show how to formulate the penalty operators such that the coupling procedure is automatically adjusted to the movements and deformations of the interface, while both stability and conservation conditions are respected.

The developed techniques are illustrated by performing numerical experiments on the linearized Euler equations and the Maxwell equations. The results corroborate the stability and accuracy of the fully discrete approximations.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2016. 38 p.
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2016:9
Keyword
Finite difference, High order accuracy, Deforming domains, Time-dependent interface, Well-posedness, Conservation, Summation-by-parts, Stability, Hyperbolic problems
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-130583 (URN)LiTH-MAT-R--2016/09--SE (ISRN)
Available from: 2016-08-17 Created: 2016-08-17 Last updated: 2016-09-27Bibliographically approved
4. A dual consistent summation-by-parts formulation for the linearized incompressible Navier-Stokes equations posed on deforming domains
Open this publication in new window or tab >>A dual consistent summation-by-parts formulation for the linearized incompressible Navier-Stokes equations posed on deforming domains
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
Incompressible Navier-Stokes equations, Deforming domain, Stability, Dual consistency, High order accuracy, Superconvergence
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-130584 (URN)LiTH-MAT-R--2016/10--SE (ISRN)
Available from: 2016-08-17 Created: 2016-08-17 Last updated: 2016-09-19Bibliographically approved
5. Summation-by-parts operators for non-simply connected domains
Open this publication in new window or tab >>Summation-by-parts operators for non-simply connected domains
2016 (English)Report (Other academic)
Abstract [en]

We construct fully discrete stable and accurate numerical schemes for solving partial differential equations posed on non-simply connected spatial domains. The schemes are constructed using summation-by-parts operators in combination with a weak imposition of initial and boundary conditions using the simultaneous approximation term technique.

In the theoretical part, we consider the two dimensional constant coefficient advection equation posed on a rectangular spatial domain with a hole. We construct the new scheme and study well-posedness and stability. Once the theoretical development is done, the technique is extended to more complex non-simply connected geometries.

Numerical experiments corroborate the theoretical results and show the applicability of the new approach and its advantages over the standard multi-block technique. Finally, an application using the linearized Euler equations for sound propagation is presented.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2016. 32 p.
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2016:11
Keyword
Initial boundary value problems, Stability, Well-posedness, Boundary conditions, Non-simply connected domains, Complex geometries
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-130585 (URN)LiTH-MAT-R--2016/11--SE (ISRN)
Available from: 2016-08-17 Created: 2016-08-17 Last updated: 2016-08-31Bibliographically approved

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