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

Gastão A. Braga, Federico C. Furtado, Long Lee

Published 2017-07-18Version 1

We introduce and systematically investigate a numerical procedure that reveals the asymptotically self-similar dynamics of solutions of partial differential equations (PDEs). The approach is based on the renormalization group (RG) theory for PDEs. This numerical version of RG method, dubbed as the numerical RG algorithm, numerically rescales the temporal and spatial variables in each iteration and drives the solutions to a fixed point exponentially fast, which corresponds to self-similar dynamics of the equations. Moreover, the procedure allows us for detecting and further finding the critical exponent of the logarithmic decay hidden in self-similar solutions. We carefully examine the ability of the algorithm for determining the critical scaling exponents in time and space that render explicitly the distinct physical effects of the solutions of the Burgers equation, depending on the initial conditions. We use the phenomena of dispersive shock waves of the Korteweg-de Vries equation to show that the algorithm can be used as a time integrator for investigating intermediate asymptotic behavior of solutions. Finally, we illustrate the ability of the numerical RG procedure for detecting and capturing the hidden logarithmic decay through a nonlinear system of cubic autocatalytic chemical reaction equations.