Consider the advection diffusion problem

<jsm>u_t+u_{x}=\nu u_{xx}+f</jsm>, <jsm> t\ge0,</jsm> and <jsm>-1 <x< 1</jsm> and <jsm>f\equiv 1</jsm>

with initial condition

<jsm>u(x,0)=\cos(\pi/2 x)</jsm>

and boundary conditions

<jsm>u(-1,t)=0</jsm> for <jsm> t\ge0,</jsm>

and

<jsm>u(1,t)=0</jsm> for <jsm> t\ge0.</jsm>

Implement a finite difference discretization for this problem. Use different values of <jsm>\nu</jsm>, say <jsm>\nu=0.1</jsm>, <jsm>\nu=0.01</jsm> <jsm>\nu=0.001</jsm>. What do you observe?