Block Spins for Partial Differential Equations

TL;DR

This paper explores a renormalisation group approach to numerically solve PDEs by coarse-graining, showing it can outperform traditional methods especially with large grid spacing and no lattice artifacts.

Contribution

It introduces a novel coarse-graining technique using renormalisation group methods to solve PDEs, calculating the perfect Laplacian operator for improved numerical accuracy.

Findings

RG method outperforms traditional discretisations with large grid spacing

No detectable lattice artifacts when a UV cutoff exists

Effective for various 1+1D PDEs with different smoothness levels

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.