Numerical Renormalization Group Algorithms for Self-Similar Asymptotics of Partial Differential Equations

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 proceedings [G. A. Braga, F. Furtado, and V. Isaia, in Proceedings of the Fifth International Conference on Dynamical Systems and Differential Equations, Pomona, CA, 2004, pp. 1--13]. A numerical version of the RG method, dubbed nRG, 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 exact or asymptotic solutions of model equations found in the 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 light on long time self-similar dynamics of certain physical systems modeled by PDEs.