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EGN 5456 Homework 7 Fall 1997
- 1.
- Show that the one-way-wave, or convection, equation
ut + a ux = f
is properly posed.
- 2.
- Determine whether the following equation is properly posed:
ut = uxx + ux + 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
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. Assume that the important cost is the
numerical work done, in other words, the total number of points
computed. (So, ignore storage considerations, assume that all
points are computed serially at the same speed regardless of the
number of spatial points, ignore the work for the boundary points.
etcetera.)
- 9.
- To what power of will 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.
- What is the price you pay for this advantage? Is it a big price to
pay?
- 13.
- How would your conclusions change for the unconditionally stable
Crank-Nicholson scheme, which has an accuracy
?
- 14.
- Determine when the forward-time,
backward space (upwind) scheme for the convection equation,
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!
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'Author: Leon van Dommelen'