Going with the Flow: a Lagrangian approach to self-similar dynamics and its consequences
Proceedings of The National Academy of Sciences(2008)
摘要
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.
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关键词
discrete time,pattern formation
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