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Unformatted text preview: g c =
c1 c2 : : : cs]T , we may write (3.2.5c) with l = 2 in the form bT c = 1=2. The
vector form of (3.2.5b) is Al = c. Thus, bT Al = 1=2, which is the same as (3.2.7) with
k = 2. Beyond k = 2, the order conditions (3.2.5c) and (3.2.7) are independent.
Although conditions (3.2.5) and (3.2.7) are only necessary for a method to be of order
p, they are su cient in many cases. The actual number of conditions for a RungeKutta
method of order p are presented in Table 3.2.2 16]. These results assume that (3.2.5b)
has been satis ed.
Order, p
1 2 3 4 5 6 7 8 9 10
No. of Conds. 1 2 4 8 17 37 85 200 486 1205
Table 3.2.2: The number of conditions for a RungeKutta method of order p 16].
; ; ; ; Theorem 3.2.1. The necessary and su cient conditions for a RungeKutta method (3.2.3) to be of second order are (3.2.5c), l = 1 2, and (3.2.7), k = 2. If (3.2.5b) is
satis ed then (3.2.5), k = 1 2, are necessary and su cient for secondorder accuracy.
Proof. We require numerous Taylor's series expansions. To be...
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.
 Spring '14
 JosephE.Flaherty

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