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Summation-By-Parts in Time
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.ORCID iD: 0000-0002-7972-6183
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
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. Vol. 251, 487-499 p.
Keyword [en]
time integration, initial value problems, high order accuracy, initial value boundary problems, boundary conditions, global methods
National Category
Computational Mathematics
URN: urn:nbn:se:liu:diva-94641DOI: 10.1016/ 000322633000027OAI: diva2:633960
Available from: 2013-06-28 Created: 2013-06-28 Last updated: 2016-03-31Bibliographically 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.
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1740
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
urn:nbn:se:liu:diva-126172 (URN)10.3384/diss.diva-126172 (DOI)978-91-7685-837-0 (Print) (ISBN)
Public defence
2016-04-22, Visionen, ingång 27, B-huset, Campus Valla, Linköping, 13:15 (English)
Swedish Research Council, 2012-1689
Available from: 2016-03-31 Created: 2016-03-17 Last updated: 2016-03-31Bibliographically approved

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