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====== IAM 950 HW1 ====== **1.** Write numerical simulation for the heat eqn $u_t = \nu u_{xx}$ on a periodic domain $[-L_x/2, L_x/2]$ 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 $\nu = 2$, Lx=100, dt=1/16, Nx=128 (number of x gridpoints). 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.** Copy and revise your heat equation code so that it simulated the 1d Swift-Hohenberg equation, $u_t = (r-1) u - 2 u_{xx} - u_{xxxx} - u^3$ on a periodic domain $[0,L_x]$ using Fourier spatial discretization and Crank-Nicolson/Adams-Bashforth semi-implicit temporal discretization. Use the initial condition $u(x,0) = \cos(ax) + 0.1 \cos(2ax)$ wehre $a=2\pi/L_x$, parameter values r = 0.2, Lx = 100, Nx = 256, dt = 1/16, and integrate from t=0 to 100. Make a colorplot of $u(x,t)$ in the $x,t$ plane. It should look like this {{:gibson:teaching:spring-2014:iam950:sh.png?direct&400|Simulations of Swift-Hohenberg eqn}} **3.** Copy and revise your Swift-Hohenberg code so that it simulates the 1d Kuramoto-Sivashinsky equation, $u_t = - u_{xx} - u_{xxxx}- u u_x$.
