Unformatted text preview: an use (5.7.12a) to calculate a new step size h such that jCk ( h)k+1y(k+1)(tn)j
or, using (5.7.12c), jCk k! k+1 ra
n k+1 j : Thus,
1=(k+1) jCk k!ran k+1j : (5.7.13a) Knowing that the local error of the next (higherorder) method of a sequence of
methods is
dn = Ck+1hk+2y(k+2)(tn) + O(hk+2)
we can also calculate an approximation of y(k+2)(tn) that can be used to change orders.
Using secondorder backward di erences of the Nordsieck vector
k+2 (k+2)
r2an k+1 h y k! (tn) :
55 and dn Ck+1k!r2 an k+1: (5.7.13b) Estimates of the error of the next lowerorder method follow directly from the nexttolast entry in the Nordsieck array. Thus, suppose that the error formula for the k ; 1order
method is
dn = Ck;1hk y(k)(tn) + O(hk+1):
Then, (cf. Example 5.7.5 and (5.7.12b))
k ;1
(
an k;1 = (kh; 1)! ynk;1) and dn Ck;1(k ; 1)!han k;1: (5.7.13c) Formulas (5.7.13b,c) can be used to increase or decrease the method order by one.
These order variations can be combined with stepsize variations to produce a LMM code
capable of both step and order adjustments. Most LMM codes do this according to the
following guidelines.
1. Start the code with a rstorder method. For the Adams methods, this would
be the Eulerbackward Euler pair. This avoids the need to incorporate separate
onestep (RungeKutta) software in the code.
2. Change order before step size. Changing order is generally much more e cient
than changing step size. Order increases and decreases in unit amounts can proceed
(
(
(
by examining changes in the sequence of derivatives ynk;1), ynk), and ynk+1). These
derivatives may be computed as described for (5.7.13). If the sequence of derivatives
are decreasing in magnitude then computational reductions can be achieved by
increasing the method order from k to k + 1. On the contrary, if the sequence of
derivatives are increasing, a reduction of order may be appropriate.
3. Include heuristics to avoid increasing the order too often. One possibility is to do
at least k steps with a method...
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 Spring '14
 JosephE.Flaherty
 Numerical Analysis, The Land, yn, Tn, Numerical ordinary differential equations

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