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Efficient fully discrete summation-by-parts schemes for unsteady flow problems
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)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. Vol. 56, no 3, 951-966 p.
Keyword [en]
Summation-by-parts in time – Unsteady flow calculations – Temporal efficiency
National Category
URN: urn:nbn:se:liu:diva-123917DOI: 10.1007/s10543-015-0599-0ISI: 000382137200007OAI: diva2:893775
Available from: 2016-01-13 Created: 2016-01-13 Last updated: 2016-10-13
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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Lundquist, TomasNordström, Jan
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