EGN 5456 Homework 8 Fall 1997

1.

Solve the unsteady heat conduction in a bar with homogeneous Dirichlet boundary conditions and an initial condition in which the temperature is maximum in the center of the bar and decays linearly to zero at each end. Use the Crank-Nicholson method to do so.

2.

Project (25% of your grade): Solve the unsteady heat conduction in a plate of length $\ell$ and height h, and a heat conduction coefficient $\kappa$. Assume that the plate is initially at zero temperature, but at time t=0, the temperature of the right-hand boundary is raised to $T_{\mbox{max}}$, and the top and bottom sides to $T_{\mbox{max}} x/\ell$. Compare the numerical solution obtained by the Peaceman-Rachford method to the exact solution for various mesh size. Try to obtain an error of 0.01 $T_{\mbox{max}}$ at time $\kappa t/h^2 = 0.05$ using the minimum number of computed points. Do this both for $\ell=h$ and for $\ell=2h$.