Abstract
We investigate the use of renormalization 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 renormalization group is superior to conventional sampling-based discretizations in representing faithfully the dynamics with a large grid spacing, introducing no detectable lattice artifacts as long as there is a natural ultraviolet cutoff in the problem. We discuss limitations and open problems of this approach.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
L. P. Kadanoff, Physics 2:263–272 (1966).
K. G. Wilson, Phys. Rev. B 4:3174, 3184 (1971).
For a review, see (e.g.) M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65:851 (1993).
N. Goldenfeld and A. McKane, unpublished (1994).
E. Katz and U.-J. Wiese, preprint comp-gas/9709001.
N. D. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (Addison-Wesley, Reading, MA, 1992).
L. Chen, N. Goldenfeld, and Y. Oono, Phys. Rev. E 54:376 (1996).
L.-Y. Chen and N. Goldenfeld, Phys. Rev. E 51:5577 (1995).
P. Hasenfratz and F. Niedermayer, Nucl. Phys. B 414:785 (1994).
A. Kast and A. J. Chorin, preprint, available at http://www.lbl.gov/~anton/KastChorin.ps
J. Swift and P. C. Hohenberg, Phys. Rev. A 15:319 (1977).
A. Novick-Cohen and G. I. Sivashinsky, Physica D 20:237 (1986); see also K. R. Elder, J. D. Gunton, and N. Goldenfeld, Phys. Rev. E 56:1631 (1997) for a finite-size scaling analysis of the phase diagram.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Goldenfeld, N., McKane, A. & Hou, Q. Block Spins for Partial Differential Equations. Journal of Statistical Physics 93, 699–714 (1998). https://doi.org/10.1023/B:JOSS.0000033249.19382.d9
Issue Date:
DOI: https://doi.org/10.1023/B:JOSS.0000033249.19382.d9