EGN 5456 Homework 2 Fall 1998

1.

Show that the one-way-wave, or convection, equation

u_t + a u_x = f

is properly posed.

2.

Determine whether the following equation is properly posed:

u_t = u_xx + u_x + u + f

3.

For the equation of question 2, what can you say about the stability condition to use for corresponding finite difference schemes?

4.

The backward-time central-space (BTCS) scheme for unsteady heat conduction in a uniform bar has the form

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

Find the truncation error.

5.

Find the order of accuracy of the backward-time, central-space scheme of question 4.

6.

Determine whether the backward-time, central-space scheme of question 4 is consistent.

7.

Examine whether the backward-time, central-space scheme of question 4 is stable.

8.

Optimize the numerical error in the backward-time, central-space scheme of question 4 for given cost. Assume that the important cost is the total number of points that must be computed.

9.

To what power of $\Delta x$ will $\Delta t$ be proportional for optimum accuracy?

10.

So, does the backward scheme have a great advantage over the forward scheme with respect to accuracy for a given effort?

11.

Can you see a clear advantage of the backward scheme over the forward scheme?

12.

Do you think the backward scheme is as easy to apply as the forward scheme?

13.

How would your conclusions change for the unconditionally stable Crank-Nicholson scheme, which has an accuracy $O(\Delta x)^2+O(\Delta t)^2$?

14.

Determine when the forward-time, backward space (upwind) scheme for the convection equation,

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

is stable. Here a>0. Use the strong version of the stability condition.

15.

You developed a new finite difference scheme for fluid flows and you used it to compute the flow for a case in which an exact solution is known. Your results are in good agreement with the exact solution. Should you still find the truncation error? Discuss fully.

16.

For the hypothetical finite difference method of the previous question, should you still do the stability analysis? Discuss fully.

17.

A certain very fast finite difference scheme for a simple equation is unstable. To be able to use this fast scheme, you get the idea of taking the spatial and time steps as powers of two. With a bit of work, this allows you to perform all computations exactly. Since you no longer have to worry about round-off errors spoiling the solution, will the computations converge to the correct solution? Discuss fully!