EGN 5456 Homework 3 Fall 1998

1.

Draw the computational stencil and discuss stability for the leapfrog scheme for the heat equation:

$${u_j^{n+1}-u_j^{n-1}\over 2\Delta t} = \kappa {u_{j+1}^n - 2 u_j^n + u_{j-1}^n\over \Delta x^2}.$$

2.

Draw the computational stencil and discuss stability for the leapfrog scheme for the convection equation:

$${u_j^{n+1}-u_j^{n-1}\over 2\Delta t} + a {u_{j+1}^n - u_{j-1}^n\over 2 \Delta x} = 0.$$

What happens in the special case that the Courant number $C=a\Delta t/\Delta x=1$?

3.

Describe how you would conduct the computation using the scheme of question . In particular, how would you find time levels n=0, 1, 2, and 3?

4.

You want to find a central finite difference formula for the third order derivative u_xxx. Use central difference operators to form an approximation involving the neighboring points j, $j\pm 1$, ..., $j\pm m$. In particular, find the approximation with the `smallest stencil' (the smallest value of m). Derive the leading order truncation error for this approximation.

5.

Discuss consistency and accuracy for the leapfrog scheme for the heat equation of question . Use operators instead of a Taylor series. What is your final conclusion about this method?

6.

Discuss consistency and accuracy for the leapfrog scheme for the convection equation of question . Use operators instead of a Taylor series. What happens in the special case that the Courant number $C=a\Delta x/\Delta t=1$? What is your final conclusion about this method?

7.

Verify accuracy and stability for the Lax-Friedrichs scheme. Use the operator tables to do both.

8.

Examine the CFL restriction for the Lax-Friedrics scheme. Compare with the exact analysis of stability and consistency.

9.

Examine the CFL restriction for the backward-time, central-space scheme for the heat equation.

10.

Verify accuracy and stability for the backward-time, central-space scheme for the heat equation. Compare with the CFL restriction.