Efficient Solution of Parabolic Equations by Krylov Approximation Methods

Abstract

This paper takes a new look at numerical techniques for solving parabolic equations by the method of lines. The main motivation for the proposed approach is the possibility of exploiting a high degree of parallelism in a simple manner. The basic idea of the method is to approximate the action of the evolution operator on a given state vector by means of a projection process onto a Krylov subspace. Thus the resulting approximation consists of applying an evolution operator of very small dimension to a known vector, which is, in turn, computed accurately by exploiting high-order rational Chebyshev and Padé approximations to the exponential. Because the rational approximation is only applied to a small matrix, the only operations required with the original large matrix are matrix-by-vector multiplications and, as a result, the algorithm can easily be parallelized and vectorized. Further parallelism is introduced by expanding the rational approximations into partial fractions. Some relevant approximation and stability issues are discussed. Some numerical experiments are presented with the method and its performance is compared with a few explicit and implicit algorithms.