Renormalization Group Analysis of Nonlinear Diffusion Equations with Periodic Coefficients

TL;DR

This paper introduces a numerical Renormalization Group method to analyze the long-time asymptotic behavior of solutions to nonlinear diffusion equations with periodic coefficients, combining homogenization theory insights.

Contribution

The paper presents a novel numerical approach that effectively computes self-similar dynamics and verifies conjectures about long-term behavior in nonlinear diffusion equations with periodic coefficients.

Findings

Successful verification of a conjecture on long-time behavior

Determination of effective or renormalized diffusion coefficients

Detailed characterization of asymptotic solutions

Abstract

In this paper we present an efficient numerical approach based on the Renormalization Group method for the computation of self-similar dynamics. The latter arise, for instance, as the long-time asymptotic behavior of solutions to nonlinear parabolic partial differential equations. We illustrate the approach with the verification of a conjecture about the long-time behavior of solutions to a certain class of nonlinear diffusion equations with periodic coefficients. This conjecture is based on a mixed argument involving ideas from homogenization theory and the Renormalization Group method. Our numerical approach provides a detailed picture of the asymptotics, including the determination of the effective or renormalized diffusion coefficient.