Επαναληπτικές μέθοδοι γραμμικών και μη γραμμικών μερικών διαφορικών εξισώσεων ελλειπτικού τύπου (Master thesis)

Κοντογιάννη, Αμαλία


A large number of physical and mathematical problems can be modeled with Partial Differential Equations (PDEs). Most physical problems are accurately described by 2nd order PDEs. Although some of these equations may under certain circumstances can be solved analytically, most of them they do not have analytical solution and can only be solved numerically with the help of numerical methods. Numerical methods have been developed rapidly over the last decades due to the evolution of numerical analysis, computer science and the construction of high-speed computers. The reason is obvious if one takes into account that numerical methods require a huge number of mathematical operations that can only be performed with the help of a computer. The problem under consideration deals with the solution of elliptical partial differential equations (Laplace and Poisson PDEs) and in particular the Dirichlet boundary value problem in two spatial dimensions. To solve numerically the physical problem described in this thesis, we end up to the solution of an algebraic system of the form Ax = b, A ∈ Rn,n, b ∈ R. To derive this system we utilized two discretization methods, namely the finite difference and the finite volume methods. The obtained numerical solutions were plotted in contour graphs and the results were compared with the analytical solutions where available. Initially, a general reference and categorization of the 2nd order PDEs is taking place. The elliptical Laplace and Poisson equations were studied. We present the analytical solutions of both Laplace and Poisson PDEs for comparison with the corresponding numerical solutions. We then describe the numerical discretization methods, the finite differences and the finite volumes methods, the application of which leads to the algebraic system mentioned above. We further present the iterative numerical methods for the solution of the 2nd order elliptic partial differential equations. In the next chapter, we present three iterative methods for the solution of linear algebraic systems, such as the steepest descend method, the conjugate gradient method and the generalized method of minimal residuals. A brief mathematical description of these method follows. Emphasis is placed on the application of the three iterative methods. We also introduce a numerical method that combines the above mentioned iterative methods with Newton’s method for solving non-linear algebraic systems. The resulting system, linear or non-linear, is numerically solved by selecting one or more of the available methods and comparing them. At the end of the thesis, the developed numerical code, written in Matlab, is presented in an appendix. We deduce that all three iterative methods give trusted results. Of these three methods, the conjugate gradient method and the generalized minimal residual method are equivalent for symmetric matrices. These methods are predominantly better compared to the steepest descent method in terms of the iterations and the convergence speed. Concerning the error, methods yield results very close to the analytical solution where it is available. In summary, we dealt with: • problems that result in large-scale systems, • implementation of iterative methods in both linear and non-linear systems (Newton’s method), • comparing analytical and numerical results that have good agreement for both Laplace and Poisson PDEs and • implementation in the programming environment Matlab with a numerical code listed in the appendix of this thesis.
Institution and School/Department of submitter: Πανεπιστήμιο Ιωαννίνων. Σχολή Θετικών Επιστημών. Τμήμα Μαθηματικών
Subject classification: Γραμμικά συστήματα
Μη γραμμικά συστήματα
Keywords: Γραμμικά και μη γραμμικά συστήματα,Linear and no linear systems
URI: http://olympias.lib.uoi.gr/jspui/handle/123456789/29016
Appears in Collections:Διατριβές Μεταπτυχιακής Έρευνας (Masters)

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