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

Abstract

We present a systematic computational approach to the study of self-similar dynamics. The approach, through the use of what can be thought of as a ``dynamic pinning condition" factors out self-similarity, and yields a transformed, non-local evolution equation. 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 approach through the porous medium equation, showing how both the Barenblatt (first kind) and the Graveleau (second kind) self-similar solutions arise in this framework. We also discuss certain implications of the dynamics of the transformed equation (which we will name "MN-dynamics"); 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.

arXiv comment14 pages, 2 figure

References in corpus (3)

• From Big Crunch to Big Bang
• Coarse Stability and Bifurcation Analysis Using Stochastic Simulators: Kinetic Monte Carlo Examples
• Renormalization study of two-dimensional convergent solutions of the porous medium equation

Cited by in corpus (6)

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• Computational coarse graining of a randomly forced 1-D Burgers equation
• From molecular dynamics to coarse self-similar solutions: A simple example using equation-free computation