University of Illinois Urbana-Champaign

A Polynomial Based Iterative Method for Linear Parabolic Equations

Schaefer, Mark Johannes

This item is only available for download by members of the University of Illinois community. Students, faculty, and staff at the U of I may log in with your NetID and password to view the item. If you are trying to access an Illinois-restricted dissertation or thesis, you can request a copy through your library's Inter-Library Loan office or purchase a copy directly from ProQuest.

Permalink

https://hdl.handle.net/2142/69578

Description

Title
A Polynomial Based Iterative Method for Linear Parabolic Equations
Author(s)
Schaefer, Mark Johannes
Issue Date
1987
Department of Study
Computer Science
Discipline
Computer Science
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Computer Science
Abstract
  • A new polynomial based method (PBM) is developed to integrate multi-dimensional linear parabolic initial-boundary-value 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 matrix-vector multiplications, involving a matrix L which arises from the spatial discretization of the differential operator. To be specific, PBM integrates the semi-discrete 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 One-Dimensional (LOD) schemes, as measured by the CPU-times required on a single CPU of a CRAY X-MP/24. It is also competitive with Crank-Nicolson which we couple with two proven iterative solvers. I recommend PBM on problems which require fourth order spatial accuracy, problems whose solutions contain significant high-frequency components, and problems whose operators cannot be split conveniently in an ADI or LOD fashion (for example, problems with mixed derivatives).
Graduation Semester
1987
Type of Resource
text
Permalink
http://hdl.handle.net/2142/69578

Owning Collections

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