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
n
n
n
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
n
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 stepsize 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 constantstep 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 constantstep Adams and BDF methods.
It is similar to EPISODE.
4. VODE is a variable step, variable order code based on the variablestep Adams
and BDF formulas. It was developed by Brown et al. 5] and is an extension of
EPISODE.
57 Code
Symbol Storage/eqn.
DEABM
22
EPISODE
....18
LSODE
...17
D02CAF : : : : : : .
19
DOPRI8 ; ; ; ; ;
9
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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 Spring '14
 JosephE.Flaherty
 Numerical Analysis, The Land, yn, Tn, Numerical ordinary differential equations

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