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An Energy Stable Summation-by-parts Formulation for General Multi-block and Hybrid Meshes
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.ORCID iD: 0000-0002-7972-6183
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: urn:nbn:se:liu:diva-126158ISRN: LiTH-MAT-R--2016/03--SELibris ID: 19712321OAI: oai:DiVA.org:liu-126158DiVA: diva2:912135
Available from: 2016-03-15 Created: 2016-03-15 Last updated: 2016-11-24Bibliographically approved
In thesis
1. High order summation-by-parts methods in time and space
Open this publication in new window or tab >>High order summation-by-parts methods in time and space
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
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:nbn:se:liu:diva-126172 (URN)10.3384/diss.diva-126172 (DOI)978-91-7685-837-0 (ISBN)
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

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An Energy Stable Summation-by-parts Formulation for General Multi-block and Hybrid Meshes(363 kB)88 downloads
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Lundquist, TomasNordström, Jan

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