Block Spins for Partial Differential Equations

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.

Renormalization group and perfect operators for stochastic differential equations

“…This imposes a stricter constraint than for the CG scheme on the power spectrum of the configuration, and is the reason why CG is a better scheme. This has been tested numerically on several model dynamics [1].…”

“…The approach outlined in this paper builds upon our previous work [1] to use RG methods to integrate out the dynamics one wishes to ignore, so that numerical methods can instead focus on the appropriate scale of interest. This is not trivial because of scale interference: the nonlinear amplification of the effect of small scale dynamics, which contaminates and eventually pollutes the large scale dynamics.…”

“…We will see here that it is possible not only to coarse-grain individual operators, as in ref. [1] but also to coarse-grain at the level of the governing differential equation. This leads to a theory that is non-local in space and time.…”

“…In addition to the work in ref. [1], there have been two attempts [9,10] to solve differential equations using perfect operators. As we will see below, it is not enough to perfectly coarse-grain the individual operators appearing in a partial differential equation: once there is a non-infinitesimal time step, coarse-graining introduces memory effects, so that the entire differential equation must be represented as coarse-grained in space-time.…”

“…The form (1) is only appropriate if κ 0 is infinite; if κ 0 < ∞, additional noise terms are generated which reflect the reduction in the number of degrees of freedom in the system. As already stressed in our earlier paper [1], it is inconsistent to work with a perfect operator with κ 0 < ∞ and to use the κ 0 = ∞ form (1) as some authors [9,10] have done. We also see no reason why coarse-grained equations should be derived by varying a coarse-grained action in the absense of a small parameter, which is the starting point of these authors.…”