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====== IAM 950 HW1  ======

**1.** Write numerical simulation for the heat eqn $u_t = \nu u_{xx}$ on a periodic domain $[0,L]$ using
Fourier spatial discretization and Crank-Nicolson temporal discretization. Verify that your code works 
correctly by simulating the Gaussian-decay solution 

<latex>
u(x,t) = (4 \pi \nu t)^{-1/2}  e^{-x^2/4\nu t}
</latex>

from $t = 1$ to $100$ for parameters $\nu = 2$, $L=100$, $dt = 1/16$, and $512$ gridpoints. 
That is, initialize your numerical code with $u(x,1)$ and then compare the results of the
numerical time integration with $u(x,t)$ evaluated from the above formula. I suggest plotting 
both quantities versus x at regular intervals in time. 

Make a plot of the numerical solution and the Gaussian solution versus $x$ at $t=100$ to turn in.
Are the two entirely consistent? If not, why not?

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