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Summation-by-parts in Time: the Second Derivative
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.
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.
LiTH-MAT-R, ISSN 0348-2960 ; 2014:11
Keyword [en]
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
URN: urn:nbn:se:liu:diva-110245ISRN: LiTH-MAT-R--2014/11--SEOAI: diva2:743661
Available from: 2014-09-04 Created: 2014-09-04 Last updated: 2016-03-31
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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