Unformatted text preview: s a collocation method is useful in many situations.
Let us illustrate one result. Theorem 3.3.4. A RungeKutta method with distinct ci , i = 1 2 : : : s, and of order at least s is a collocation method satisfying (3.3.12), (3.3.15) if and only if it satis es
the order conditions
s
X q 1 cq
aij cj = i
i q = 1 2 : : : s:
(3.3.16)
; j =1 q Remark 1. The order conditions (3.3.15) are related to the previous conditions (3.2.5c,
3.2.7) (cf. 16], Section II.7). 30 Proof. We use the Lagrange interpolating polynomial (3.3.13) to represent any polynomial P ( ) of degree s ; 1 as
s
X
P ( ) = P (cj )Lj ( ):
j =1 Regarding P ( ) as u (tn + h), integrate to obtain
Zc
Zc
s
X
u(tn 1 + cih) ; yn 1 = P ( )d = P (cj ) Lj ( )d
0 i ; ; i j =1 0 0 Assuming that (3.3.15a) is satis ed, we have
Zc
s
X
P ( )d = aij P (cj )
i i = 1 2 : : : s: j =1 0 i = 1 2 : : : s: Now choose P ( ) = q 1, q = 1 2 : : : s, to obtain (3.3.16). The proof of the converse
follows the same arguments (cf. 16], Section II.7).
; Now, we might...
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 Spring '14
 JosephE.Flaherty
 Numerical Analysis, yn, Tn, Numerical ordinary differential equations

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