uu.seUppsala University Publications

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High Order Finite Difference Methods in Space and TimePrimeFaces.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. 28
##### Series

Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1104-232X ; 880
##### Keywords [en]

finite difference methods, Navier-Stokes equations, high order time discretization, deferred correction, stability
##### National Category

Computational Mathematics
##### Research subject

Numerical Analysis
##### Identifiers

URN: urn:nbn:se:uu:diva-3559ISBN: 91-554-5721-5 (print)OAI: oai:DiVA.org:uu-3559DiVA, id: diva2:163250
##### Public defence

2003-10-24, Room 1211, Polacksbacken, Uppsala University, Uppsala, 10:15 (English)
##### Opponent

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##### Supervisors

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#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt831",{id:"formSmash:j_idt831",widgetVar:"widget_formSmash_j_idt831",multiple:true}); Available from: 2003-09-24 Created: 2003-09-24 Last updated: 2011-10-26Bibliographically approved
##### List of papers

In this thesis, high order accurate discretization schemes for partial differential equations are investigated.

In the first paper, the linearized two-dimensional Navier-Stokes equations are considered. A special formulation of the boundary conditions is used and estimates for the solution to the continuous problem in terms of the boundary conditions are derived using a normal mode analysis. Similar estimates are achieved for the discretized equations. For the discretization, a second order finite difference scheme on a staggered mesh is used. In Paper II, the analysis for the second order scheme is used to develop a fourth order scheme for the fully nonlinear Navier-Stokes equations. The fully nonlinear incompressible Navier-Stokes equations in two space dimensions are considered on an orthogonal curvilinear grid. Numerical tests are performed with a fourth order accurate Padé type spatial finite difference scheme and a semi-implicit BDF2 scheme in time.

In Papers III-V, a class of high order accurate time-discretization schemes based on the deferred correction principle is investigated. The deferred correction principle is based on iteratively eliminating lower order terms in the local truncation error, using previously calculated solutions, in each iteration obtaining more accurate solutions. It is proven that the schemes are unconditionally stable and stability estimates are given using the energy method. Error estimates and smoothness requirements are derived. Special attention is given to the implementation of the boundary conditions for PDE. The scheme is applied to a series of numerical problems, confirming the theoretical results.

In the sixth paper, a time-compact fourth order accurate time discretization for the one- and two-dimensional wave equation is considered. Unconditional stability is established and fourth order accuracy is numerically verified. The scheme is applied to a two-dimensional wave propagation problem with discontinuous coefficients.

1. Boundary conditions and estimates for the linearized Navier-Stokes equations on staggered grids$(function(){PrimeFaces.cw("OverlayPanel","overlay68906",{id:"formSmash:j_idt925:0:j_idt935",widgetVar:"overlay68906",target:"formSmash:j_idt925:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. A Compact Higher Order Finite Difference Method for the Incompressible Navier-Stokes Equations$(function(){PrimeFaces.cw("OverlayPanel","overlay76024",{id:"formSmash:j_idt925:1:j_idt935",widgetVar:"overlay76024",target:"formSmash:j_idt925:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. Deferred Correction Methods for Initial Value Problems$(function(){PrimeFaces.cw("OverlayPanel","overlay68489",{id:"formSmash:j_idt925:2:j_idt935",widgetVar:"overlay68489",target:"formSmash:j_idt925:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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5. Error Estimates for Deferred Correction Methods in Time$(function(){PrimeFaces.cw("OverlayPanel","overlay76026",{id:"formSmash:j_idt925:4:j_idt935",widgetVar:"overlay76026",target:"formSmash:j_idt925:4:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

6. A Compact Fourth Order Time Discretization Method for the Wave Equation$(function(){PrimeFaces.cw("OverlayPanel","overlay76027",{id:"formSmash:j_idt925:5:j_idt935",widgetVar:"overlay76027",target:"formSmash:j_idt925:5:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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