Unformatted text preview: . Hairer and Wanner 13], Section V.5, compare several codes for a
suite of sti problems. We report their results in Figures 5.7.3 and 5.7.4. The codes used
for this test follow.
1. LSODE and VODE are the codes described previously with the BDF option set.
2. DEBDF is a driver for LSODE for sti systems.
3. SPRINT, developed by Berzins and Furzeland 2], contains several multistep methods and solution packages. The one used for the test is a blended multistep method
that is Astable to order four.
4. SECDER and ROBER are multistep codes that won't be described further.
5. LADAMS is LSODE with the Adams methods selected.
6. RADAU5 is a fthorder implicit RungeKutta method based on collocation at
Radau points.
The results are scattered and di cult to interpret. The implicit RungeKutta method
appears to do reasonably well on the small sti problems of Figure 5.7.3 but less well
on the larger problems of Figure 5.7.4. The Adams software did well on several of the
smaller sti problems but had di culties with the larger problems. The opposite appears
true for VODE. The SPRINT software also did well on several of the larger problems.
Some nal notes of comparison between RungeKutta and LMMs follow:
1. RungeKutta methods are preferred to Adams methods when function evaluations
are inexpensive.
2. In general, the stability of an Adams predictorcorrector pair is better than that of
an explicit RungeKutta method of the same order.
61 Figure 5.7.3: CPU time vs. error for a suite of small sti problems 13].
62 Figure 5.7.4: CPU times vs. error for a suite of large sti problems 13].
63 3. Estimation of local errors is less expensive with a LMM than with a RungeKutta
method. As seen in Examples 5.7.7 and 5.7, however, RungeKutta methods remain
competitive with LMMs as long as the function evaluations aren't too expensive.
4. Fixedorder RungeKutta methods are easier to implement than LMMs. However,
good software of both types exist.
5. For large sti problems, BDFs are much more e cient than implicit RungeKutta
m...
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 Spring '14
 JosephE.Flaherty
 Numerical Analysis, The Land, yn, Tn, Numerical ordinary differential equations

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