21 the simplest scheme j 1 is the midpoint rule the

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Unformatted text preview: ith sti IVPs, Radau schemes should be suitable for singularly perturbed BVPs. This will determine the proper choice of the xed endpoint, as described in Chapter 10. The pointwise accuracy of a J point Radau scheme is O(h2 1). J J J J; 3. Lobatto points. The points 1 = 0, = 1, and the remaining points , k = 2 3 : : : J ; 1, are selected as the roots of the Legendre polynomial of degree J ; 2. The simplest scheme (J = 2) is the trapezoidal rule. The pointwise accuracy of a J point scheme is O(h2( 1) ). J k J; Example 9.1.1 ( 1], Chapter 5). As in Example 9.1.1, consider 8 y + y = ( 8 ; x2 )2 0<x<1 y (0) = y(1) = 0: x Ascher et al. 1] solve this problem using 0 00 0 two-point Gauss-Legendre collocation at 1 1 1 = 2 (1 ; p ) 3 2= 1 (1 + p ) 1 2 3 three-point Lobatto collocation at 1 =0 2 = 1=2 and 17 3=1 N 2-Point Gauss 3-Point Lobatto 2 0:20 10 3 0:17 10 4 5 5 0:64 10 0:57 10 6 10 0:46 10 6 0:37 10 7 20 0:33 10 7 0:23 10 8 8 40 0:23 10 0:15 10 9 80 0:16 10 9 0:91 10 11 3-Point Gauss 0:14 10 6 0:7...
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.

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