%%
% We set up the semidiscretization and initial condition in x just as
% before.
m = 100;  
[x, Dx, Dxx] = diffper(m, [0,1]);
u0 = exp( -60*(x-0.5).^2 );

%%
% Now, though, we apply ode45 to the initial-value problem % u'=D_xx u
tfinal = 0.05;
ODE = @(t,u) Dxx*u;  
[t, U] = ode45(ODE, [0,tfinal], u0);
U = U';    % rows for space, columns for time
figure(1)
plot(x, U(:,1:400:2001))
xlabel('x')
ylabel('u(x,t)')
legend(num2str(t(1:400:2001)))
title('Heat equation by ode45')
grid on

%%
% The solution is at least plausible. But the number of time steps that
% were selected automatically is surprisingly large, considering how
% smoothly the solution changes.
num_steps_ode45 = length(t)-1

%%
% Now we apply another solver, ode15s. 
[t, U] = ode15s(ODE, [0,tfinal], u0);
U = U';    % rows for space, columns for time
figure(2)
plot(x, U(:,1:10:41))
xlabel('x')
ylabel('u(x,t)')
legend(num2str(t(1:10:41)))
title('Heat equation by ode15s')
grid on

%% 
% The number of steps selected is reduced by a factor of more than 50!
num_steps_ode15s = length(t)-1