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Error analysis of summation-by-parts formulations: Dispersion, transmission and accuracyPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2017 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Linköping: Linköping University Electronic Press, 2017. , p. 27
##### Series

Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1886
##### National Category

Computational Mathematics Mathematical Analysis Control Engineering Fluid Mechanics and Acoustics
##### Identifiers

URN: urn:nbn:se:liu:diva-143059DOI: 10.3384/diss.diva-143059ISBN: 978-91-7685-427-3 (print)OAI: oai:DiVA.org:liu-143059DiVA, id: diva2:1158370
##### Public defence

2017-12-12, Ada Lovelace,, B-huset, Campus Valla, Linköping, 13:15 (English)
##### Opponent

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#####

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Available from: 2017-11-20 Created: 2017-11-20 Last updated: 2017-11-20Bibliographically approved
##### List of papers

In this thesis we consider errors arising from finite difference operators on summation-by-parts (SBP) form, used in the discretisation of partial differential equations. The SBP operators are augmented with simultaneous-approximation-terms (SATs) to weakly impose boundary conditions. The SBP-SAT framework combines high order of accuracy with a systematic construction of provably stable boundary procedures, which renders it suitable for a wide range of problems.

The first part of the thesis treats wave propagation problems discretised using SBP operators on coarse grids. Unless special care is taken, inaccurate approximations of the underlying dispersion relation materialises in the form of an incorrect propagation speed. We present a procedure for constructing SBP operators with minimal dispersion error. Experiments indicate that they outperform higher order non-optimal SBP operators for flow problems involving high frequencies and long simulation times.

In the second part of the thesis, the formal order of accuracy of SBP operators near boundaries is analysed. We prove that the order in the interior of a diagonal norm based SBP operator must be at least twice that of the boundary stencil, irrespective of the grid point distribution near the boundary. This generalises the classical theory posed on uniform and conforming grids. We further show that for a common class of SBP operators, the diagonal norm defines a quadrature rule of the same order as the interior stencil. Again, this result is independent of the grid.

In the final contribution if the thesis, we introduce the notion of a transmission problem to describe a general class of problems where different dynamics are coupled in time. Well-posedness and stability analyses are performed for continuous and discrete problems. A general condition is obtained that is necessary and sufficient for the transmission problem to satisfy an energy estimate. The theory provides insights into the coupling of fluid flow models, multi-block formulations, numerical filters, interpolation and multi-grid implementations.

1. Uniformly Best Wavenumber Approximations by Spatial Central Difference Operators$(function(){PrimeFaces.cw("OverlayPanel","overlay849400",{id:"formSmash:j_idt1164:0:j_idt1168",widgetVar:"overlay849400",target:"formSmash:j_idt1164:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. A simple and efficient incompressible Navier-Stokes solver for unsteady complex geometry flows on truncated domains$(function(){PrimeFaces.cw("OverlayPanel","overlay1089226",{id:"formSmash:j_idt1164:1:j_idt1168",widgetVar:"overlay1089226",target:"formSmash:j_idt1164:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. Summation-by-Parts Operators with Minimal Dispersion Error for Coarse Grid Flow Calculations$(function(){PrimeFaces.cw("OverlayPanel","overlay1088493",{id:"formSmash:j_idt1164:2:j_idt1168",widgetVar:"overlay1088493",target:"formSmash:j_idt1164:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. On the order of Accuracy of Finite Difference Operators on Diagonal Norm Based Summation-By-Parts Form$(function(){PrimeFaces.cw("OverlayPanel","overlay1140552",{id:"formSmash:j_idt1164:3:j_idt1168",widgetVar:"overlay1140552",target:"formSmash:j_idt1164:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

5. Well-posed and Stable Transmission Problems$(function(){PrimeFaces.cw("OverlayPanel","overlay1153062",{id:"formSmash:j_idt1164:4:j_idt1168",widgetVar:"overlay1153062",target:"formSmash:j_idt1164:4:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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