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Unformatted text preview: of order k before considering an order change.
56 4. Starting values when changing step size or order may be computed by a Taylor's
series when using the Nordsieck representation (5.7.10).
There are also variable step size LMMs ( 12], Section III.5). The coe cients of these
formulas are functions of h. The variable step formulas are generally more stable than
the uniform step formulas but are less e cient.
Some available LMM codes are
1. DEABM is a modi cation of a code developed by Shampine and Gordon 16]. This
code implements a variable step size divided di erence representation of the Adams
formulas. It uses a PECE strategy and includes order variation. Let's go over the
order variation scheme to illustrate the technique. After performing a step with
an order k method, compute estimates dk;2, dk;1, and dk of the local error of
solutions with methods of order k ; 2, k ; 1, and k, respectively. Reduce the order
to k ; 1 if
max kdk;2k kdk;1k] kdk jj:
n (5.7.14) Increase the order when a step is successful, (5.7.14) is violated, and a constant
step size is used. (Remember, these are variable step-size methods.) Estimates of
local discretization errors are obtained using approaches similar to (5.7.13). Norms
are used for vector systems.
2. EPISODE, developed by Byrne and Hindmarsh 7], is a variable step, variable order
implementation of the constant-step Adams and BDF methods using the Nordsieck
representation. For nonsti problems, functional iteration uses a P (EC ) strategy.
Newton's method is used for sti problems.
3. LSODE is another implementation of the constant-step Adams and BDF methods.
It is similar to EPISODE.
4. VODE is a variable step, variable order code based on the variable-step Adams
and BDF formulas. It was developed by Brown et al. 5] and is an extension of
D02CAF : : : : : : .
DOPRI8 ; ; ; ; ;
Table 5.7.1: Legends for Figures 5.7.1 and 5.7.2 and code storage 12].
5. DASSL, devel...
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