Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
High order summation-by-parts methods in time and space
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis develops the methodology for solving initial boundary value problems with the use of summation-by-parts discretizations. The combination of high orders of accuracy and a systematic approach to construct provably stable boundary and interface procedures makes this methodology especially suitable for scientific computations with high demands on efficiency and robustness. Most classes of high order methods can be applied in a way that satisfies a summation-by-parts rule. These include, but are not limited to, finite difference, spectral and nodal discontinuous Galerkin methods.

In the first part of this thesis, the summation-by-parts methodology is extended to the time domain, enabling fully discrete formulations with superior stability properties. The resulting time discretization technique is closely related to fully implicit Runge-Kutta methods, and may alternatively be formulated as either a global method or as a family of multi-stage methods. Both first and second order derivatives in time are considered. In the latter case also including mixed initial and boundary conditions (i.e. conditions involving derivatives in both space and time).

The second part of the thesis deals with summation-by-parts discretizations on multi-block and hybrid meshes. A new formulation of general multi-block couplings in several dimensions is presented and analyzed. It collects all multi-block, multi-element and  hybrid summation-by-parts schemes into a single compact framework. The new framework includes a generalized description of non-conforming interfaces based on so called summation-by-parts preserving interpolation operators, for which a new theoretical accuracy result is presented.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2016. , 21 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1740
Keyword [en]
summation-by-parts, time integration, stiff problems, weak initial conditions, high order methods, simultaneous-approximation-term, finite difference, discontinuous Galerkin, spectral methods, conservation, energy stability, complex geometries, non-conforming grid interfaces, interpolation
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-126172DOI: 10.3384/diss.diva-126172ISBN: 978-91-7685-837-0 (print)OAI: oai:DiVA.org:liu-126172DiVA: diva2:912667
Public defence
2016-04-22, Visionen, ingång 27, B-huset, Campus Valla, Linköping, 13:15 (English)
Opponent
Supervisors
Funder
Swedish Research Council, 2012-1689
Available from: 2016-03-31 Created: 2016-03-17 Last updated: 2016-03-31Bibliographically approved
List of papers
1. Summation-By-Parts in Time
Open this publication in new window or tab >>Summation-By-Parts in Time
2013 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 251, 487-499 p.Article in journal (Refereed) Published
Abstract [en]

We develop a new high order accurate time-integration technique for initial value problems. We focus on problems that originate from a space approximation using high order finite difference methods on summation-by-parts form with weak boundary conditions, and extend that technique to the time-domain. The new time-integration method is global, high order accurate, unconditionally stable and together with the approximation in space, it generates optimally sharp fully discrete energy estimates. In particular, it is shown how stable fully discrete high order accurate approximations of the Maxwells’ equations, the elastic wave equations and the linearized Euler and Navier-Stokes equations can obtained. Even though we focus on finite difference approximations, we stress that the methodology is completely general and suitable for all semi-discrete energy-stable approximations. Numerical experiments show that the new technique is very accurate and has limited order reduction for stiff problems.

Place, publisher, year, edition, pages
Elsevier, 2013
Keyword
time integration, initial value problems, high order accuracy, initial value boundary problems, boundary conditions, global methods
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-94641 (URN)10.1016/j.jcp.2013.05.042 (DOI)000322633000027 ()
Available from: 2013-06-28 Created: 2013-06-28 Last updated: 2017-12-06Bibliographically approved
2. The SBP-SAT Technique for Initial Value Problems
Open this publication in new window or tab >>The SBP-SAT Technique for Initial Value Problems
2014 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 270, 86-104 p.Article in journal (Refereed) Published
Abstract [en]

A detailed account of the stability and accuracy properties of the SBP-SAT technique for numerical time integration is presented. We show how the technique can be used to formulate both global and multi-stage methods with high order of accuracy for both stiff and non-stiff problems. Linear and non- linear stability results, including A-stability, L-stability and B-stability are proven using the energy method for general initial value problems. Numerical experiments corroborate the theoretical properties.

Place, publisher, year, edition, pages
Elsevier, 2014
Keyword
time integration, initial value problems, high order accuracy, initial boundary value problems, boundary conditions, global methods, stability, convergence, summation-by-parts operators, stiff problems
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-105802 (URN)10.1016/j.jcp.2014.03.048 (DOI)000336406200006 ()
Available from: 2014-04-07 Created: 2014-04-07 Last updated: 2017-12-05
3. Efficient fully discrete summation-by-parts schemes for unsteady flow problems
Open this publication in new window or tab >>Efficient fully discrete summation-by-parts schemes for unsteady flow problems
2016 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 56, no 3, 951-966 p.Article in journal (Refereed) Published
Abstract [en]

We make an initial investigation into the temporal efficiency of a fully discrete summation-by-parts approach for unsteady flows. As a model problem for the Navier–Stokes equations we consider a two-dimensional advection–diffusion problem with a boundary layer. The problem is discretized in space using finite difference approximations on summation-by-parts form together with weak boundary conditions, leading to optimal stability estimates. For the time integration part we consider various forms of high order summation-by-parts operators and compare with an existing popular fourth order diagonally implicit Runge–Kutta method. To solve the resulting fully discrete equation system, we employ a multi-grid scheme with dual time stepping.

Place, publisher, year, edition, pages
Springer, 2016
Keyword
Summation-by-parts in time – Unsteady flow calculations – Temporal efficiency
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-123917 (URN)10.1007/s10543-015-0599-0 (DOI)000382137200007 ()
Available from: 2016-01-13 Created: 2016-01-13 Last updated: 2017-11-30
4. Summation-by-parts in Time: the Second Derivative
Open this publication in new window or tab >>Summation-by-parts in Time: the Second Derivative
2016 (English)Report (Other academic)
Abstract [en]

A new technique for time integration of initial value problems involving second derivatives is presented. The technique is based on summation-by-parts operators and weak initial conditions and lead to optimally sharp energy estimates. The schemes obtained in this way use wide operators, are unconditionally stable and high order accurate. The additional complications when using compact operators in time are discussed in detail and it is concluded that the existing compact formulations designed for space approximations are not appropriate. As an application we focus on the wave equation and derive optimal fully discrete energy estimates which lead to unconditional stability. The scheme utilizes wide stencil operators in time, whereas the spatial operators can have both wide and compact stencils. Numerical calculations verify the stability and accuracy of the new methodology.

Place, publisher, year, edition, pages
Linköping University Electronic Press, 2016. 27 p.
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2014:11
Keyword
Time integration, second derivative approximations, initial value problems, high order accuracy, initial value boundary problems, boundary conditions, stability, convergence, summation-by-parts operators
National Category
Mathematics Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-110245 (URN)LiTH-MAT-R--2014/11--SE (ISRN)
Available from: 2014-09-04 Created: 2014-09-04 Last updated: 2016-03-31
5. On the Suboptimal Accuracy of Summation-by-parts Schemes with Non-conforming Block Interfaces
Open this publication in new window or tab >>On the Suboptimal Accuracy of Summation-by-parts Schemes with Non-conforming Block Interfaces
2016 (English)Report (Other academic)
Abstract [en]

We derive a bound on the formal accuracy of interpolation schemes for energy stable summation-by-parts discretizations on non-conforming multiblock grids. This result explains the suboptimal accuracy for such schemes reported in previous works. Numerical simulations confirm a corresponding reduced convergence rate in both maximum and L2 norms.

Place, publisher, year, edition, pages
Linköping University Electronic Press, 2016. 12 p.
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2015:16
Keyword
Summation-by-parts, High order finite difference methods, Multi-block discretizations, Interpolation operators.
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-122816 (URN)LiTH-MAT-R--2015/16--SE (ISRN)
Available from: 2015-11-25 Created: 2015-11-25 Last updated: 2016-09-26Bibliographically approved
6. An Energy Stable Summation-by-parts Formulation for General Multi-block and Hybrid Meshes
Open this publication in new window or tab >>An Energy Stable Summation-by-parts Formulation for General Multi-block and Hybrid Meshes
2016 (English)Report (Other academic)
Abstract [en]

Most high order methods for solving conservation laws can be shown to satisfy a summation-by-parts rule. In this work we present a general framework for multi-block and multi-element summation-by-parts implementations in several dimensions that includes most, if not all of the previously known schemes on summation-by-parts form. This includes finite volume, spectral and nodal discontinuous Galerkin methods, as well as high order multi-block finite difference schemes on curvilinear domains. We use the framework to derive general conditions for conservation and stability, and formulate extended representations of conservative and energy stable couplings between completely general multi-block, multi-element or hybrid meshes.

Place, publisher, year, edition, pages
Linköping University Electronic Press, 2016. 35 p.
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2016:03
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-126158 (URN)LiTH-MAT-R--2016/03--SE (ISRN)
Available from: 2016-03-15 Created: 2016-03-15 Last updated: 2016-11-24Bibliographically approved

Open Access in DiVA

fulltext(382 kB)143 downloads
File information
File name FULLTEXT01.pdfFile size 382 kBChecksum SHA-512
5753ba8dff962a5575aa0001ad49f08a5eacf8ae150ca922aa7f9d4366c2b8d1a3dd79101850844773a53c8cf078623b94a785b5b81d9f0bd4abd3cd37574029
Type fulltextMimetype application/pdf
omslag(77 kB)4 downloads
File information
File name COVER01.pdfFile size 77 kBChecksum SHA-512
485a31ef16d57ad16db909671d8f21d5f4982283fae52962d457063fc735ef8304e595eaaf3a5f4082c03af6192759c5f41e86a0eea75b04f32ea3438ec7cb64
Type coverMimetype application/pdf

Other links

Publisher's full text

Search in DiVA

By author/editor
Lundquist, Tomas
By organisation
Computational MathematicsFaculty of Science & Engineering
Computational Mathematics

Search outside of DiVA

GoogleGoogle Scholar
Total: 143 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

doi
isbn
urn-nbn

Altmetric score

doi
isbn
urn-nbn
Total: 3516 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf