# 313a using 3313c z utn 1 h yn 1 h u tn 1 hd

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Unformatted text preview: s a collocation method is useful in many situations. Let us illustrate one result. Theorem 3.3.4. A Runge-Kutta 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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