An Allen-Cahn equation with continuation

Nick Trefethen, November 2010

The Allen-Cahn equation is a reaction-diffusion that arises in material science: u_tt = Eps*u"+u-u^3=, where Eps is a small parameter. Here as an ODE boundary-value problem we shall consider a steady-state version of this problem on the interval [0,10] with a sinusouidal forcing term:

Eps*u" + u - u^3 = sin(x), u(0) = 1, u(10) = -1.

If we try a very small value of Eps without a well-chosen initial guess, Chebfun will not converge. Instead let's begin by solving the problem with the rather large initial guess Eps = 1:

Eps = 1;
dom = [0,10];
x = chebfun('x',dom);
f = sin(x);
cheboppref('plotting',0.01)
N = chebop(@(u) Eps*diff(u,2)+u-u.^3,dom,1,-1);
tic, u = N\f; t = toc;

LW = 'linewidth'; lw = 2; FS = 'fontsize'; fs = 14;
close, plot(u,LW,lw)
s = 'Eps = %5.1e    length(u) = %d    time = %3.1f secs';
title(sprintf(s,Eps,length(u),t),FS,fs)

We now progressively reduce Eps to get sharper and sharper solutions. We use a simple continuation method, in which the initial guess for each iteration is the previous solution.

Epsvec = [.5 .2 .1 .03 .009 .003];
for j = 1:length(Epsvec)
  close all
  Eps = Epsvec(j);
  N = chebop(@(u) Eps*diff(u,2)+u-u.^3,dom,1,-1);
  N.guess = u;
  tic, u = N\f; t = toc;
  close, plot(u,LW,lw)
  title(sprintf(s,Eps,length(u),t),FS,fs), snapnow
end