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This is a group (pair) assignment. You will investigate the numerical
solution of the 1D heat equation and report your observations and results.
You should compare three numerical schemes: 1) the forward difference scheme
as displayed in class, 2) the Richardson and 3) the Crank-Nicolson schemes,
as found on page 248 of the text.
- Determine the stability of schemes 2) and 3) [Ex 6.3.12].
- Run each scheme for at least five different initial conditions.
(Include the initial condition in Figure 6.3.4. Vary the mesh size and
observe what happens. Compare to any known exact solutions.
- Use what you consider the best scheme and for several initial
conditions, solve the heat equation for
- u(0,t)=0 and u(1,t)=1.
- x=0 insulated and u(1,t)=0.
All work should be typed with double-spacing and 12 pt font.
You will be expected to use correct English grammar and punctuation. You
will be graded on the evidence of work, mathematical detail and
understanding, proper exposition and neatness. Your work should also be
supported with properly labeled and embedded Matlab plots. Any references
used should be cited as well. These reports will count 100 points towards
the homework component of your grade. All Matlab code should be provided in
one of your appendices in the report.
Format of report:
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Summary of Results (Abstract)
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Discussion of Schemes used and your stability
results. Put your determination of these results in an appendix.
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Discuss your numerical results, the initial
conditions used and what you found out. For example:
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What happened when you changed the mesh?
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Is there evidence supporting the reported
truncation errors for the schemes?
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Does the numerical work support your stability
conditions?
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Anything else you thought to look at?
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Conclusion - What did you learn? Which schemes would
you use in the future? Did you find any surprises?
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Appendices: Place derivations, code, etc in a set of
appendices. The paper should read smoothly so that the reader is not
distracted by unnecessary detail, unless you think there is something that
needs to be spelled out to clarify your discussion.
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