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The six problems used for testing are described in

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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 A-stable 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 fth-order implicit Runge-Kutta method based on collocation at Radau points. The results are scattered and di cult to interpret. The implicit Runge-Kutta 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 Runge-Kutta and LMMs follow: 1. Runge-Kutta methods are preferred to Adams methods when function evaluations are inexpensive. 2. In general, the stability of an Adams predictor-corrector pair is better than that of an explicit Runge-Kutta 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 Runge-Kutta method. As seen in Examples 5.7.7 and 5.7, however, Runge-Kutta methods remain competitive with LMMs as long as the function evaluations aren't too expensive. 4. Fixed-order Runge-Kutta 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 Runge-Kutta m...
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