Abstract

. In this paper we take 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 Pad'e 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 ...

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