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Unformatted text preview: (3.4.1b). Then, yn ! y(tn)
for all t 2 0 T ] as h ! 0 and N ! 1. Now, zn , de ned by (3.4.5), is identical to yn,
so zn must also converge to y(tn). Additionally, we have proven that zn converges to
the solution z(t) of the IVP (3.4.4). Uniqueness of the solutions of (3.4.4) and (3.4.1b)
imply that z(t) = y(t). This is impossible unless the consistency condition (3.4.3) is
Global error bounds for general one-step methods (3.4.1) have the same form that we
saw in Chapter 2 for Euler's method. Thus, a method of order p will converge globally
as O(hp). Theorem 3.4.3. Let satisfy the conditions of Theorem 3.4.2 and let the one-step
method be of order p. Then, the global error en = y(tn) ; yn is bounded by hp
jenj CL (eLT ; 1): (3.4.11) Proof. Since the one-step method is of order p, there exists a positive constant C such
that the local error dn satis es jdnj C hp+1: The remainder of the proof follows the lines of Theorem 2.1.1. Problems
1. Prove Theorem 3.4.3. 3.5 Implementation: Erro...
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- Spring '14