PD&C HW2pD

Problem Statement: 1. Express the heat diffusion equation from C1 as an algebraic equation via Crank-Nicolson differencing. 2. Using von Neumann stability analysis, show that the Crank-Nicolson differencing scheme is stable regardless of the choice for r= . 3. Simulate the time dependent response of a 50 cm aluminum bar (thermal diffusivity = 1e-4 m2/s) uniform bar initially at 20C after its ends are raised then maintained at 50 and 80C respectively. Discretize the bar into 1 cm sections and solve the equations from D1 using a time step of 0.1 s, scipy.linalg.solve_banded for 10000 time steps. Plot the simulated results in 20 time step intervals. 4. Repeat D3 for a time step of 10 seconds, i.e., r= = 5, and show that while the simulation at early times if inaccurate, the numerical solution remains stable.