Numerical renormalization group algorithms for self-similar solutions of partial differential equations

TL;DR

This paper analyzes a numerical renormalization group method for PDEs that efficiently identifies self-similar solutions, validating its accuracy and demonstrating its broad applicability to complex physical models.

Contribution

It provides a detailed validation of the numerical RG algorithm and showcases its versatility across various PDEs for studying long-term self-similar behavior.

Findings

The numerical RG algorithm converges exponentially to self-similar solutions.

Validation against exact and asymptotic solutions confirms accuracy.

Applicable to a wide range of PDEs with complex long-term dynamics.

Abstract

We systematically study a numerical procedure that reveals the asymptotically self-similar dynamics of solutions of partial differential equations (PDEs). This procedure, based on the renormalization group (RG) theory for PDEs, appeared initially in a conference proceeding by Braga et al. \cite{BFI04}. This numerical version of RG method, dubbed as the numerical RG (nRG) algorithm, numerically rescales the temporal and spatial variables in each iteration and drives the solutions to a fixed point exponentially fast, which corresponds to the self-similar dynamics of the equations. In this paper, we carefully examine and validate this class of algorithms by comparing the numerical solutions with either the exact or the asymptotic solutions of the model equations in literature. The other contribution of the current paper is that we present several examples to demonstrate that this class of nRG algorithms can be applied to a wide range of PDEs to shed lights on longtime self-similar dynamics of certain physical models that are difficult to analyze, both numerically and analytically.