The shallow water equations on the unit sphere with scattered data

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Copyright: Tregubov, Ilya
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Abstract
This dissertation deals with different aspects of the shallow water equations (SWEs) on the unit sphere in the three dimensional Euclidean space. Such equations arise by vertically integrating the Navier--Stokes equations and the mass conservation equation. They are a simplification of the equations describing various meteorological phenomena such as tides in coastal areas. The SWEs are used as a primary test problem for new efficient methods for real life hydrodynamical problems. In this dissertation, firstly, we prove the existence of weak solutions for the SWEs in consideration by the Galerkin method. We prove that the Galerkin approximations converge to the solution of the original system. When solving global meteorological problems, data are often collected by satellites. Therefore it becomes inconvenient to use methods that require structured grids. Consequently, the development of finite element methods that have less geometric requirements for mesh generation becomes more and more popular. A second contribution of this thesis is the design of a finite element method using spherical splines in the case scattered data are involved in the modelling of the SWEs. We implement our proposed method to the standard Williamson test set for the inviscid SWEs with satellite data. To generate more realistic flows it is important to include viscosity in the model. As a third contribution, we derive {\it a priori} error estimates for this method using spherical splines, when viscosity is included in both the momentum and continuity equations. We also calculate estimated orders of convergence that support our theoretical results. As is well known, the matrices arising when using finite element methods are often ill conditioned. To overcome the ill conditioness, as a fourth contribution of the thesis, we study two different types of preconditioners, namely the additive Schwarz and the alternate triangular preconditioners. We calculate numerical bounds for the condition numbers to verify the effectiveness of both preconditioners for the spherical SWEs problem.
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Author(s)
Tregubov, Ilya
Supervisor(s)
Tran, Thanh
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Publication Year
2014
Resource Type
Thesis
Degree Type
PhD Doctorate
UNSW Faculty
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