Unformatted text preview: 1)
3
3 Y2 = yn 1 + h f (tn 1 + h Y1) + f (tn Y2)]
2
3
yn = yn 1 + h 3f (tn 1 + h Y1) + f (tn Y2)]:
4
3
We can check by constructing a Taylor's series that this method is indeed third order.
Hairer et al. 16], Section II.2, additionally show that our necessary conditions for thirdorder accuracy are also su cient in this case.
The computation of Y1 can be recognized as the backward Euler method for onethird
of the time step h. The computation of Y2 and yn are not recognizable in terms of simple
quadrature rules. Since the method is thirdorder, its local error is O(h4).
We can also construct an SDIRK method by insisting that a11 = a22 . Enforcing this
condition and using the previous relations gives two methods having the tableau
; ; ; ; ; ; 1; 1;2
1=2 where 0
1=2 1
= 1 (1 p ):
2
3
p
The method with = (1 + 1= 3)=2 is Astable while the other method has a bounded
stability region. Thus, this would be the method of choice.
28 Let us conclude this Section by noting a relationship between implicit R...
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 Spring '14
 JosephE.Flaherty
 Numerical Analysis, yn, Tn, Numerical ordinary differential equations

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