Renormalization group calculation of asymptotically self-similar dynamics

Pages: 131 - 141, Issue Special, August 2005

 Abstract        Full Text (209.6K)              

G. A. Braga - Departamento de Matemática, Universidade Federal de Minas Gerais, Caixa Postal 1621, Belo Horizonte, 30161-970, Brazil (email)
Frederico Furtado - Department of Mathematics, University of Wyoming, Laramie, 82071, United States (email)
Vincenzo Isaia - Department of Mathematics, Alfred State College, NY, United States (email)

Abstract: We present a systematic numerical procedure for the computation of asymptotically self-similar dynamics of physical systems whose evolution is modeled by PDEs. This approach is based on the renormalization group (RG) for PDEs, which was originally introduced by N. Goldenfeld, Y. Oono and collaborators, and was further developed by J. Bricmont, A. Kupiainen and collaborators. We explain how successive iterations of a discrete RG transformation in space and time drive the system towards a fixed point, which corresponds to a self-similar dynamics. The iteration of the RG transformation renders explicit the relative importance of the distinct physical effects being modeled in the long-time dynamics. The resulting nu- merical procedure is very efficient and provides a detailed picture of the asymptotics, including scaling exponents, profile functions, and prefactors. We illustrate the ef- fectiveness of the procedure on a set of examples of nonlinear PDEs, including cases where nonlinear effects are asymptotically irrelevant or neutral. In the latter case the asymptotic scaling laws obeyed by the dynamics frequently contain logarithmic corrections, which are detected and successfully handled by the RG procedure.

Keywords:  Renormalization group, self-similar dynamics, asymptotic behavior, partial differential equations.
Mathematics Subject Classification:  Primary: 35K55, 35B40, 35B33, 35B27; Secondary:74S20.

Received: September 2004;      Revised: April 2005;      Published: September 2005.