A polynomial based iterative method for linear parabolic equations

摘要

A new polynomial based method (PBM) is developed to integrate multi-dimensional linear parabolic initial-boundary value problems. It is based on L2-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 recommended 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 ut = L(t)u + b(t),u,b∈Rn and L∈Rn×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 in the numerical examples considered, it compares favorably to standard methods from the classes of Alternating Direction Implicit and Locally One-Dimensional 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 is coupled with two proven iterative solvers. I recommend PBM on problems which require high spatial accuracy and problems whose solutions contain significant high-frequency components.