EGN 5456 Homework 5 Fall 1998

1.

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 $T_{\mbox{max}} x/\ell$.

• Read the values of $\ell$, h, $\kappa$, and $T_{\mbox{max}}$ from an input file, along with the values of $j_{\mbox{max}}$ and $l_{\mbox{max}}$ and the parameter that determines the time step. In the input file, specify the values to be $\ell=2$, h=1, $\kappa=3$, $T_{\mbox{max}}=2$, and try various values for $j_{\mbox{max}}$, selecting $l_{\mbox{max}}$ to give square mesh cells. In the input file, also put the number of desired output times $n_{\mbox{out}}$ followed by the output times $t_{\mbox{max}}$.
• Write a subroutine that computes the exact solution.
• Write a Peaceman-Rachford scheme to compute a finite difference solution.
• Optimize the ratio of mesh size to time step for a given computational effort (= points computed) to give best results for a given effort.
• Create a Mitchell-Fairweather scheme from the Peaceman-Rachford one.
• Optimize the ratio of square mesh size to time step for the same computational effort as for the Peaceman-Rachford scheme to give the best results.
• See whether the Mitchwell-Fairweather scheme gives better results than the Peaceman-Rachford
• Increase the effort by a factor 10 and compare again.

Contour lines of constant temperature are a good way to plot the temperature evolution. Otherwise you might want to plot the temperature evolution at x=1 and the one at $y=\frac12$.

The program must meet the neatness recommendations of the handout or significant credit (30%) will be lost.