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====== IAM 950 HW1 ====== **1.** Write numerical simulation for the heat eqn $u_t = \nu u_{xx}$ on a periodic domain $[-L/2,L/2]$ 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 128 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? **2.** Write a numerical simulation for the 1d Swift-Hohenberg equation, $u_t = (r-1) u - u_{xx} - u_{xxxx}$ on a periodic domain $[-L/2, L/2]$ using Fourier spatial discretization and Crank-Nicolson/Adams-Bashforth semi-implicit temporal discretization. The above heat equation code is a good starting point. **3.** Write a numerical simulation for the 1d Kuramoto-Sivashinsky equation, $u_t = - u_{xx} - u_{xxxx}- u u_x$ on a periodic domain $[-L/2, L/2]$ using Fourier spatial discretization and Crank-Nicolson/Adams-Bashforth semi-implicit temporal discretization. This should be a very minor modification of the above Swift-Hohenberg code.
