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The quadrature discretization method and its applications

University of British Columbia

A discretization method referred to as the quadrature discretization method is introduced and studied in this thesis. The quadrature discretization method is a spectral method based on a grid of points that coincide with the points of a quadrature. The quadrature is based on a set of nonclassical polynomials orthogonal with respect to some weight function over some interval. The method is flexible with respect to the choice of the weight function and quadrature points so that the optimum accuracy and convergence in the solution of differential equations and/or partial differential equations can be obtained. The properties of the quadrature discretization method are studied and compared with classical spectral methods as well as finite difference methods. Several analytic model problems in one and three dimension are studied. The quadrature discretization method competes well with classical spectral methods and is far more superior than the finite difference method. In some cases it provides significant improvement in the accuracy and convergence of the solution of the problem over other methods. The main objective of the quadrature discretization method is to determine the weight function that defines the polynomial basis set and hence the grid points that provide optimum convergence in a given application. The quadrature discretization method is applied to a large class of time dependent Fokker-Planck equations. Several weight functions are used and the results are compared with several other methods. The weight functions that have often provided rapid convergence of the eigenvalues and eigenfunctions of the Fokker-Planck operator are the steady solutions at infinite time. The quadrature discretization method is also employed in the solution of Schrodinger equations. The weight functions that are used are related to the ground state wave functions if known, or some approximate form. The eigenvalues and eigenfunctions of four different potential functions discussed extensively in the literature are calculated and compared with the published values. The eigen-problem of a two-dimensional Schrodinger equation with the Henon-Heiles potential is also calculated with the quadrature discretization method. The rate of convergence of the eigenvalues and eigenfunctions of the Schrodinger equations is very rapid with this approach.

Thesis/Dissertation

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2009-06-19

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http://hdl.handle.net/2429/9493

Mathematics

University of British Columbia

UBCV

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