Nicoli Matteo, Misbah Chaouqi, Politi Paolo
Physique de la Matière Condensée, École Polytechnique, CNRS, Palaiseau, F-91128, France.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Jun;87(6):063302. doi: 10.1103/PhysRevE.87.063302. Epub 2013 Jun 7.
Many nonlinear partial differential equations (PDEs) display a coarsening dynamics, i.e., an emerging pattern whose typical length scale L increases with time. The so-called coarsening exponent n characterizes the time dependence of the scale of the pattern, L(t)≈t(n), and coarsening dynamics can be described by a diffusion equation for the phase of the pattern. By means of a multiscale analysis we are able to find the analytical expression of such diffusion equations. Here, we propose a recipe to implement numerically the determination of D(λ), the phase diffusion coefficient, as a function of the wavelength λ of the base steady state u(0)(x). D carries all information about coarsening dynamics and, through the relation |D(L)|=/~L(2)/t, it allows us to determine the coarsening exponent. The main conceptual message is that the coarsening exponent is determined without solving a time-dependent equation, but only by inspecting the periodic steady-state solutions. This provides a much faster strategy than a orward time-dependent calculation. We discuss our method for several different PDEs, both conserved and not conserved.
许多非线性偏微分方程（PDEs）呈现出粗化动力学，即一种典型长度尺度(L)随时间增加的新兴模式。所谓的粗化指数(n)表征了模式尺度的时间依赖性，(L(t)≈t^n)，并且粗化动力学可以用模式相位的扩散方程来描述。通过多尺度分析，我们能够找到此类扩散方程的解析表达式。在此，我们提出一种方法，用于数值实现确定相位扩散系数(D(λ))，它是基态稳态(u^{(0)}(x))波长(λ)的函数。(D)携带了关于粗化动力学的所有信息，并且通过关系(|D(L)|≈L^2/t)，它使我们能够确定粗化指数。主要的概念要点是，粗化指数不是通过求解含时间的方程来确定，而是仅通过考察周期性稳态解来确定。这提供了一种比正向含时间计算快得多的策略。我们针对几种不同的PDEs（包括守恒型和非守恒型）讨论我们的方法。
