A Polynomial Based Iterative Method for Linear Parabolic Equations

88 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1987.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).U of I OnlyRestricted to the U of I community idenfinitely during batch ingest of legacy ETD