Block Spins for Partial Differential Equations

Details Block Spins for Partial Differential Equations Block Spins for Partial Differential Equations

1998-02-10

arxiv.org/abs/cond-mat/9802103v1

Physics

Condensed Matter

Statistical Mechanics

8 pages, RevTeX, 8 figures, contribution to L.P. Kadanoff festschrift (J. Stat. Phys)

Scientific paper

10.1023/B:JOSS.0000033249.19382.

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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