Solution diverges for 1D heat equation using Crank-Nicholson

I am trying to solve the 1D heat equation using the Crank-Nicholson method. I have managed to code up the method but my solution blows up. I'm using Neumann conditions at the ends and it was advised that I take a reduced matrix and use that to find the interior points and then afterwards. Would this work? My code is blowing up and it should work but for some reason it isn't and I can't see the reason why. Any suggestions? Code below:

%This program is meant to test out the Crank-Nicolson scheme using a simple
%nonlinear diffusion scheme.
n=5000;m=30;
t=linspace(0,20,n);% set time and distance scales
x=linspace(0,1,m);
dx=x(2)-x(1);dt=t(2)-t(1); %Increments
s=dt/(2*dx^2);%Useful for the solution.
u=zeros(n,m); %set up solution matrix
p=s*ones(1,m-1); q=-(1+2*s)*ones(1,m);
Z=diag(p,1)+diag(p,-1)+diag(q);
A=Z(2:m-1,2:m-1);
%Add in intial condition:
u(1,:)=exp(-5*(x-0.5).^2);
v=zeros(m-2,1);
%Solve the system
for i=2:n-1
  %Construct the RHS for the solution
  for j=2:m-1
      v(j-1)=s*u(i-1,j+1)-(2*s+1)*u(i-1,j)-s*u(i-1,j-1);
  end
  %Solve for the new time step
  w=A\v;
  u(i,2:m-1)=w;
  u(i,1)=u(i,2);
  u(i,end)=u(i,end-1);
end

Answer 1

[Torsten]

%This program is meant to test out the Crank-Nicolson scheme using a simple
%nonlinear diffusion scheme.
n=5000;m=150;
t=linspace(0,10,n);% set time and distance scales
x=linspace(0,1,m);
dx=x(2)-x(1);dt=t(2)-t(1); %Increments
s=dt/(2*dx^2);%Useful for the solution.
u=zeros(n,m); %set up solution matrix
A=zeros(m,m);
A(1,1) = 1;
A(1,2) = -1;
for j=2:m-1
  A(j,j-1) = -s;
  A(j,j) = (1+2*s);
  A(j,j+1) = -s;
end
A(m,m-1)= -1;
A(m,m) = 1;
%Add in intial condition:
u(1,:)=exp(-5*(x-0.5).^2);
v=zeros(m,1);
%Solve the system
for i=2:n-1
  %Construct the RHS for the solution
  for j=2:m-1
      v(j)=s*u(i-1,j+1)+(1-2*s)*u(i-1,j)+s*u(i-1,j-1);
  end
  %Solve for the new time step
  w=A\v;
  u(i,:)=w;
end
end

[Matthew Hunt]

Why was my method of shoving in the vectors not working then?

And how did you insert the boundary conditions?

[Torsten]

I invested the time to understand your code. Now it's up to you to invest the time to see the differences to mine.

[Matthew Hunt]

Actually, I saw the error in my code which still exists in the code you gave me. The problem wasn't in the allocation of the matrix A which you seemed to think but was in the definition of v which you didn't change.

[Torsten]

The definition of v has changed - take a closer look.