EGN 5456 Homework 4 Fall 1998
Numerical Solution of Heat Conduction in a Bar

1.

Fix up your program of homework 1 first so that it meets the good programming requirements.

2.

Solve the same problem as in homework 1, but this time, use the Crank-Nicholson method.

3.

Verify the unconditional stability.

4.

Verify that the numerical error is second order in both space and time.

5.

To what power of the round-off error is the numerical effort proportional?

6.

How does this compare with the explicit scheme at $\kappa\Delta t /(\Delta x)^2 = {1\over 2}$?

7.

How does this compare with the explicit scheme at $\kappa\Delta t /(\Delta x)^2 = {1\over 6}$?

8.

Convert your program to the modified Crank-Nicholson method and address all above questions again.

9.

Discuss the CFL condition for the following beloved scheme

$${s^{n+1}_j - s^n_j\over \Delta t} + {u\over 2} {s^{n+1}_j -... ...r\Delta x} + {u\over 2} {s^n_{j+1} - s^n_j\over\Delta x} = 0$$

10.

Why is the CFL condition prediction of the previous question wrong?

11.

Discuss the CFL condition for the Du Fort-Frankel scheme.

12.

Is the CFL condition a condition for stability for the Du Fort Frankel scheme?