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31 at the collocation points we have e a e 0 ij ij ij

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Unformatted text preview: expect w(x) to have p ; J bounded derivatives. Thus, expand w(x) in a Taylor's series of the form w(x) = P 1(x) + O(h ) p;J p;J ; i where P 1(x) is a polynomial of degree p ; J ; 1. If the one-step method is accurate to order p then p;J ; Z xj xj Since ;1 P p; J ; J j =1 (x ; )dx = 0: ij Y J j we have 1(x) Y Z xj xj ;1 =1 (x ; ) = O(h ) J ij G( x)Le(x)dx = O(h P ;J i i )O(h )h J i i 2 (x = 1 j; x ): j (9.3.15a) The result (9.3.13a) is obtained by summing the above relation over the subintervals. When 2 (x 1 x ) then we are only able to show that Z j; j xj xj ;1 G( x)Le(x)dx = O(h +1) J i Summing (9.3.15a,b) yields (9.3.13b). Superconvergence occurs whenever p > J + 1. 24 2 (x 1 j; x ): j (9.3.15b) Bibliography 1] U.M. Ascher, R. Mattheij, and R. Russell. Numerical Solution of Boundary Value Problems for Ordinary Di erential Equations. SIAM, Philadelphia, second edition, 1995. 2] C. de Boor. A Practical Guide to Splines. Springer-Verlag, New York, 1978. 3] C. de Boor and B. Swartz. Collocation at gaussian points. SIAM J. Numer. Anal., 10:582{687, 1973. 4] J.E. Flaherty and W. Mathon. Collocation with polynomial and tension splines for singularly perturbed boundary value problems. SIAM J. Sci. Stat. Comput, 1:260{ 289, 1980. 5] E. Isaacson and H.B. Keller. Analysis of Numerical Methods. John Wiley and Sons, New York, 1966. 25...
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