p34.m

% p34.m - Allen-Cahn eq. u_t = u_xx + u - u^3, u(-1)=-1, u(1)=1
% (compare p6.m and p32.m)
% Differentiation matrix and initial data:
N = 20; [D,x] = cheb(N); D2 = D^2;
% use full-size matrix
D2([1 N+1],:) = zeros(2,N+1);
% for convenience
eps = 0.01; dt = min([.01,50*N^(-4)/eps]);
t = 0; v = .53*x + .47*sin(-1.5*pi*x);
% Solve PDE by Euler formula and plot results:
tmax = 100; tplot = 2; nplots = round(tmax/tplot);
plotgap = round(tplot/dt); dt = tplot/plotgap;
xx = -1:.025:1; vv = polyval(polyfit(x,v,N),xx);
plotdata = [vv; zeros(nplots,length(xx))]; tdata = t;
for i = 1:nplots
for n = 1:plotgap
t = t+dt; v = v + dt*(eps*D2*(v-x) + v - v.^3); % Euler
end
vv = polyval(polyfit(x,v,N),xx);
plotdata(i+1,:) = vv; tdata = [tdata; t];
end
clf, subplot('position',[.1 .4 .8 .5])
mesh(xx,tdata,plotdata), grid on, axis([-1 1 0 tmax -1 1]),
view(-60,55), colormap([0 0 0]), xlabel x, ylabel t, zlabel u