Going with the Flow: a Lagrangian approach to self-similar dynamics and its consequences

We present a systematic computational approach to the study of self-similar dynamics. Through the use of what can be thought of as a “dynamic pinning condition” self-similarity is factored out, and a transformed, non-local evolution equation is obtained. The approach, which is capable of treating both first and second kind self-similar solutions, yields as a byproduct the self-similarity exponents of the original dynamics. We illustrate the procedure through the porous medium equation, showing how both the Barenblatt (first kind) and the Graveleau (second kind) self-similar solutions naturally arise in this framework. We also discuss certain implications of the dynamics of the transformed equation; in particular we discuss the discrete-time implementation of the approach, and connections with time-stepper based methods for the “coarse” integration/bifurcation analysis of microscopic simulators.