**This is an old revision of the document!** ----
====== 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?
