Files in this item
Files  Description  Format 

application/pdf 8721754.pdf (3MB)  (no description provided) 
Description
Title:  A Polynomial Based Iterative Method for Linear Parabolic Equations 
Author(s):  Schaefer, Mark Johannes 
Department / Program:  Computer Science 
Discipline:  Computer Science 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Computer Science 
Abstract:  A new polynomial based method (PBM) is developed to integrate multidimensional linear parabolic initialboundaryvalue problems. It is based on $L\sb2$approximations to $f(z) = (1  exp(z))/z,f(0) = 1,$ over ellipses in the complex plane using expansions of f in Chebychev polynomials. The calculation of the Fourier Coefficients requires numerical integration over only a single line segment in the complex plane whose length and orientation depend on the step size and the parabolic operator itself. The simplicity with which these coefficients are obtained rests on special properties of the Chebychev polynomials. Most of the work in PBM consists of matrixvector multiplications, involving a matrix L which arises from the spatial discretization of the differential operator. To be specific, PBM integrates the semidiscrete problem $u\sb{t} = L(t)u + b(t), u,b$ in $R\sp{n}$ and L in $R\sp{n\times n},$ and requires only a modest amount of storage (a few vectors of order n). Due to the analyticity of f it has good convergence properties and compares favorably to other standard methods from the classes of Hopscotch, Alternating Direction Implicit (ADI) and Locally OneDimensional (LOD) schemes, as measured by the CPUtimes required on a single CPU of a CRAY XMP/24. It is also competitive with CrankNicolson which we couple with two proven iterative solvers. I recommend PBM on problems which require fourth order spatial accuracy, problems whose solutions contain significant highfrequency components, and problems whose operators cannot be split conveniently in an ADI or LOD fashion (for example, problems with mixed derivatives). 
Issue Date:  1987 
Type:  Text 
Description:  88 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1987. 
URI:  http://hdl.handle.net/2142/69578 
Other Identifier(s):  (UMI)AAI8721754 
Date Available in IDEALS:  20141215 
Date Deposited:  1987 
This item appears in the following Collection(s)

Dissertations and Theses  Computer Science
Dissertations and Theses from the Dept. of Computer Science 
Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois