Adaptive, Fast and Oblivious Convolution in Evolution Equations with Memory

Details Adaptive, Fast and Oblivious Convolution in Evolution Equations with Memory Adaptive, Fast and Oblivious Convolution in Evolution Equations with Memory

2006-11-07

arxiv.org/abs/math/0611168v1

Mathematics

Numerical Analysis

22 pages

Scientific paper

To approximate convolutions which occur in evolution equations with memory terms, a variable-stepsize algorithm is presented for which advancing N steps requires only O(N log(N)) operations and O(log(N)) active memory, in place of O(N^2) operations and O(N) memory for a direct implementation. A basic feature of the fast algorithm is the reduction, via contour integral representations, to differential equations which are solved numerically with adaptive step sizes. Rather than the kernel itself, its Laplace transform is used in the algorithm. The algorithm is illustrated on three examples: a blow-up example originating from a Schr\"odinger equation with concentrated nonlinearity, chemical reactions with inhibited diffusion, and viscoelasticity with a fractional order constitutive law.

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