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It remains to determine the collocation points that

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Unformatted text preview: d is a cubic polynomial it can be evaluated exactly by the two-point Gauss-Legendre quadrature formula (cf. 5], Chapter 7) Z1 1 ; 1 1 f ( )d = f (; p ) + f ( p ) + E 3 (9.1.31a) 3 where the discretization error E is given by 1 E = 135 f ( ) 2 (;1 1): iv (9.1.31b) Thus, applying (9.1.31) to (9.1.30), we have 1 1 1 1 1 1 (; p + 2 1 )(; p ; 2 2)P (; p ) + ( p + 2 1)( p ; 2 2)P ( p ) = 0: 3 3 3 3 3 3 Once again, the integrand in (9.1.30) is a cubic polynomial so E = 0. We see that we can satisfy the above condition by choosing 1 1 = 2= p : 23 (9.1.32a) Expressed in terms of the original variables through (9.1.29b), the collocation points are i 1=x 12; i; = h p i 23 i 2=x 12+ i; = h: p i 23 (9.1.32b) Regardless of how the result is expressed, the key is to perform collocation at the GaussLegendre points mapped to the appropriate interval. 11 N Midpoint Rule Collocation 2 0:20 10 3 5 0:64 10 5 10 0:10 10 4 0:46 10 6 20 0:26 10 5 0:33 10 7 40 0:65 10 6 0:23 10 8 80 0:16 10 6 0:16 10 9 ; ; ; ; ; ; ; ; ; ; Table 9.1.1: Maximum pointwise errors in the s...
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