Coarsening dynamics in one dimension: The phase diffusion equation and its numerical implementation

TL;DR

This paper develops a numerical method to determine the coarsening exponent in nonlinear PDEs by analyzing steady-state solutions, offering a faster alternative to time-dependent simulations.

Contribution

It introduces a novel numerical approach to compute the phase diffusion coefficient from steady states, enabling efficient prediction of coarsening dynamics in various PDEs.

Findings

The method accurately predicts coarsening exponents across different PDEs.

It significantly reduces computational effort compared to traditional time-dependent simulations.

The approach is applicable to both conserved and non-conserved PDEs.

Abstract

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, $L(t)\approx t^n$, 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 $D(\lambda)$, the phase diffusion coefficient, as a function of the wavelength $\lambda$ of the base steady state $u_0(x)$. $D$ carries all information about coarsening dynamics and, through the relation $|D(L)| \simeq L^2 /t$, 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 forward time-dependent calculation. We discuss our method for several different PDEs, both conserved and not conserved.