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

Abstract

To approximate convolutions which occur in evolution equations with memory terms, a variable-step-size 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 blowup example originating from a Schrödinger equation with concentrated nonlinearity, chemical reactions with inhibited diffusion, and viscoelasticity with a fractional order constitutive law.