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Stable computation of differentiation matrices and scattered node stencils based on Gaussian radial basis functions
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
2012 (English)Report (Other academic)
Place, publisher, year, edition, pages
2012.
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-020
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-179191OAI: oai:DiVA.org:uu-179191DiVA: diva2:543636
Available from: 2012-08-13 Created: 2012-08-09 Last updated: 2012-09-12Bibliographically approved
In thesis
1. High Order Local Radial Basis Function Methods for Atmospheric Flow Simulations
Open this publication in new window or tab >>High Order Local Radial Basis Function Methods for Atmospheric Flow Simulations
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Since the introduction of modern computers, numerical methods for atmospheric simulations have routinely been applied for weather prediction, and in the last fifty years, there has been a steady improvement in the accuracy of forecasts. Accurate numerical models of the atmosphere are also becoming more important as researchers rely on global climate simulations to assess and understand the impact of global warming.

The choice of grid in a numerical model is an important design decision and no obvious optimal choice exists for computations in spherical geometry. Despite this disadvantage, grid-based methods are found in all current circulation models. A different approach to the issue of discretizing the surface of the sphere is given by meshless methods, of which radial basis function (RBF) methods are becoming prevalent.

In this thesis, RBF methods for simulation of atmospheric flows are explored. Several techniques are introduced to increase the efficiency of the methods. By utilizing a novel algorithm for adaptively placing the node points, accuracy is shown to improve by over one order of magnitude for two relevant test problems. The computational cost can also be reduced by using a local finite difference-like RBF scheme. However, this requires a stabilization mechanism for the hyperbolic problems of interest here. A hyper-viscosity scheme is introduced to address this issue.

Another stability issue arising from the ill-conditioning of the RBF basis for almost-flat basis functions is also discussed in the thesis, and two algorithms are proposed for dealing with this stability problem. The algorithms are specifically tailored for the task of creating finite difference weights using RBFs and are expected to overcome the issue of stationary error in local RBF collocation.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2012. 37 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 951
Keyword
Radial basis function, meshless method, high order finite difference, atmospheric flow, hyper-viscosity, stable algorithm
National Category
Computational Mathematics
Research subject
Scientific Computing
Identifiers
urn:nbn:se:uu:diva-179194 (URN)978-91-554-8421-7 (ISBN)
Public defence
2012-09-21, Room 2446, Polacksbacken, Lägerhyddsvägen 2D, Uppsala, 10:00 (English)
Opponent
Supervisors
Available from: 2012-08-29 Created: 2012-08-09 Last updated: 2013-01-22Bibliographically approved

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http://www.it.uu.se/research/publications/reports/2012-020/

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Larsson, ElisabethLehto, Erik

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