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

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)?tn, 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(?), the phase diffusion coefficient, as a function of the wavelength ? of the base steady state u0(x). D carries all information about coarsening dynamics and, through the relation |D(L)|?L2/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.