Block spins for partial differential equations

Nigel Goldenfeld, Alan McKane, Qing Hou

    Research output: Contribution to journalArticlepeer-review


    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 smooth-ness 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.
    Original languageEnglish
    Pages (from-to)699-714
    Number of pages15
    JournalJournal of Statistical Physics
    Issue number3-4
    Publication statusPublished - Nov 1998


    • Numerical analysis
    • Partial differential equations
    • Pattern formation
    • Renormalization group
    • Spatiotemporal chaos


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