Adaptive, Fast and Oblivious Convolution in Evolution Equations with Memory
Please always quote using this URN: urn:nbn:de:0297-zib-9399
- 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 Schrödinger equation with concentrated nonlinearity, chemical reactions with inhibited diffusion, and viscoelasticity with a fractional order constitutive law.
Author: | Maria Lopez-Fernandez, Christian Lubich, Achim Schädle |
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Document Type: | ZIB-Report |
Tag: | Volterra integral equations; adaptivity; anomalous diffusion; convolution quadrature; fractional order visco; numerical inverse Laplace transform |
MSC-Classification: | 65-XX NUMERICAL ANALYSIS / 65Mxx Partial differential equations, initial value and time-dependent initial- boundary value problems / 65M99 None of the above, but in this section |
65-XX NUMERICAL ANALYSIS / 65Rxx Integral equations, integral transforms / 65R20 Integral equations | |
Date of first Publication: | 2006/11/02 |
Series (Serial Number): | ZIB-Report (06-45) |
ZIB-Reportnumber: | 06-45 |
Published in: | Appeared in: SIAM Journal on Scientific Computing Vol.30, No.2 (2008) |
DOI: | https://doi.org/10.1137/060674168 |