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# Thus letting yx denote the approximate solution we

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Unformatted text preview: ubinterval endpoints are not collocation points. The rst ve Legendre polynomials are shown in Table 9.2.1. The simplest scheme (J = 1) is the midpoint rule. The pointwise accuracy of a J -point scheme is O(h2 ). k J J 2. Radau points. As described in Section 3.3, either 1 = 0 or = 1 and the remaining points , k = 2 3 : : : J , or k = 1 2 : : : J ; 1, respectively, are selected to achieve maximal order. This involves selecting the points on ;1 1] as the roots of the Radau polynomial of degree J J k R ( ) = P ( ) 2J J 1 P 1( ) ; J J J; 16 J P() 0 1 2 3 4 1 J j 1 (3 2 ; 1) 2 2 2 (5 ; 3) 1 (35 4 ; 30 2 + 3) 8 , j = 1 2 ::: J 0 1 3 p 0, q3 5 0:3399810436, 0:8611363116 Table 9.2.1: Legendre polynomials P ( ) and their roots for J = 0 1 : : : 4. J where P ( ) is the Legendre polynomial of degree J . The negative sign is chosen with = 1 and the positive sign is selected with 1 = 0. The simplest scheme (J = 1) is the forward or backward Euler method when 1 = 0 or = 1, respectively. Judging from our experience w...
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