### Solving the heat equation with non-constant diffusion

Consider the equation on [0,2*pi]

u_t = (a(x) u_x)_x

where a(x) = 1 + eps*sin(x). This is a heat equation with non-constant diffusion. Consider eps between 0 and 1. Intuitively, how do you expect the diffusion constant will affect things?

Modify your Crank-Nicolson to code up this equation. Use either the finite-difference code or the spectral code. Make sure that your code was working before you modify it. If you need a working code to cannibalize, get one from Natasha (rojkovsk@math).

If you did the spectral case, demonstrate that your code is working correctly by showing that it's O(dt^2) in time. If you did the finite difference case, demonstrate that your code is working correctly by showing that it's O(dt^2) in time and O(dx^2) in space. In both cases, check that it's right for the eps=0 case by comparing your solution to a known exact solution. You have to check the O(dt^2) and O(dx^2) stuff with eps not equal to zero. In this case you don't know the exact solution and you'll have to figure out how to check the convergence when you don't know the exact solution.

Now that your code is working, choose various initial data that interest you and run the code with different values of epsilon. Present evolution plots demonstrating that it's behaving as you expected.

### Solving the Burger's equation

On the web-page, I've provided a leap-frog code for Burger's equation. Modify this code to include diffusion.

u_t = eps u_xx -(u^2)_x/2 = L(u) + N(u)

where L is linear and N is nonlinear. Treat the diffusion in a Crank-Nicolson way:

(u_{i,j+1}-u_{i,j-1})/(2 dt) = eps/2 (u_xx)_{i,j+1} + eps/2 (u_xx)_{i,j-1} + N(u)_{i,j}

Now write a separate code where you compute Burger's equation explicitly:

(u_{i,j+1}-u_{i,j})/(dt) = eps (u_xx)_{i,j} + N(u)_{i,j}

From class, if eps is small then a time step dt and space step dx that were originally okay may become too large.

Choose initial data and run the two codes head-to-head. That is, with the same time-step, the same space-step, the same eps, and the same final time. Plot their power spectra at different times on the same plot (crank-nicolson: solid, explicit: dot-dash). Demonstrate that the explicit code can become unstable. And demonstrate the effect of this instability on the solution itself.

With my code, do a number of runs with a range of eps to demonstrate to yourself the effect of eps on the evolution of the power spectrum and on the evolution of the solution.