%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
