uu.seUppsala University Publications

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Summation-by-Parts Operators for High Order Finite Difference MethodsPrimeFaces.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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2003 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

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

Uppsala: Acta Universitatis Upsaliensis , 2003. , p. 23
##### Series

Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1104-232X ; 828
##### Keywords [en]

finite difference methods, accuracy, stability, dissipation
##### National Category

Computational Mathematics
##### Research subject

Numerical Analysis
##### Identifiers

URN: urn:nbn:se:uu:diva-3434ISBN: 91-554-5596-4 (print)OAI: oai:DiVA.org:uu-3434DiVA, id: diva2:162794
##### Public defence

2003-05-09, Room 2146, Polacksbacken, Uppsala University, Uppsala, 10:15 (English)
##### Opponent

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##### Supervisors

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#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt462",{id:"formSmash:j_idt462",widgetVar:"widget_formSmash_j_idt462",multiple:true}); Available from: 2003-04-17 Created: 2003-04-17 Last updated: 2011-10-27Bibliographically approved
##### List of papers

High order accurate finite difference methods for hyperbolic and parabolic initial boundary value problems (IBVPs) are considered. Particular focus is on time dependent wave propagating problems in complex domains. Typical applications are acoustic and electromagnetic wave propagation and fluid dynamics. To solve such problems efficiently a strictly stable, high order accurate method is required.

Our recipe to obtain such schemes is to: i) Approximate the (first and second) derivatives of the IBVPs with central finite difference operators, that satisfy a summation by parts (SBP) formula. ii) Use specific procedures for implementation of boundary conditions, that preserve the SBP property. iii) Add artificial dissipation. iv) Employ a multi block structure.

Stable schemes for weakly nonlinear IBVPs require artificial dissipation to absorb the energy of the unresolved modes. This led to the construction of accurate and efficient artificial dissipation operators of SBP type, that preserve the energy and error estimate of the original problem.

To solve problems on complex geometries, the computational domain is broken up into a number of smooth and structured meshes, in a multi block fashion. A stable and high order accurate approximation is obtained by discretizing each subdomain using SBP operators and using the Simultaneous Approximation Term (SAT) procedure for both the (external) boundary and the (internal) interface conditions.

Steady and transient aerodynamic calculations around an airfoil were performed, where the first derivative SBP operators and the new artificial dissipation operators were combined to construct high order accurate upwind schemes. The computations showed that for time dependent problems and fine structures, high order methods are necessary to accurately compute the solution, on reasonably fine grids.

The construction of high order accurate SBP operators for the second derivative is one of the considerations in this thesis. It was shown that the second derivative operators could be closed with two order less accuracy at the boundaries and still yield design order of accuracy, if an energy estimate could be obtained.

1. Boundary Procedures for Summation-by-Parts Operators$(function(){PrimeFaces.cw("OverlayPanel","overlay108482",{id:"formSmash:j_idt511:0:j_idt515",widgetVar:"overlay108482",target:"formSmash:j_idt511:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. Stable and Accurate Artificial Dissipation$(function(){PrimeFaces.cw("OverlayPanel","overlay99063",{id:"formSmash:j_idt511:1:j_idt515",widgetVar:"overlay99063",target:"formSmash:j_idt511:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. Accuracy requirements for transient aerodynamics$(function(){PrimeFaces.cw("OverlayPanel","overlay76191",{id:"formSmash:j_idt511:2:j_idt515",widgetVar:"overlay76191",target:"formSmash:j_idt511:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. Finite Difference Approximations of Second Derivatives on Summation by Parts Form$(function(){PrimeFaces.cw("OverlayPanel","overlay76934",{id:"formSmash:j_idt511:3:j_idt515",widgetVar:"overlay76934",target:"formSmash:j_idt511:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

5. Steady State Computations Using Summation-by-Parts Operators$(function(){PrimeFaces.cw("OverlayPanel","overlay76946",{id:"formSmash:j_idt511:4:j_idt515",widgetVar:"overlay76946",target:"formSmash:j_idt511:4:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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