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Block Spins for Partial Differential Equations

Nigel Goldenfeld·Alan McKane·Qing Hou
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Abstract

We investigate the use of renormalisation group methods to solve partial differential equations (PDEs) numerically. Our approach focuses on coarse-graining the underlying continuum process as opposed to the conventional numerical analysis method of sampling it. We calculate exactly the coarse-grained or `perfect' Laplacian operator and investigate the numerical effectiveness of the technique on a series of 1+1-dimensional PDEs with varying levels of smoothness in the dynamics: the diffusion equation, the time-dependent Ginzburg-Landau equation, the Swift-Hohenberg equation and the damped Kuramoto-Sivashinsky equation. We find that the renormalisation group is superior to conventional sampling-based discretisations in representing faithfully the dynamics with a large grid spacing, introducing no detectable lattice artifacts as long as there is a natural ultra-violet cut off in the problem. We discuss limitations and open problems of this approach.

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BibTeXRIS

Nigel Goldenfeld, Alan McKane, Qing Hou. 1998-02-10. Block Spins for Partial Differential Equations. https://doi.org/10.1023/b%3Ajoss.0000033249.19382.d9

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