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1 we recall that when 1 there are boundary layers

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Unformatted text preview: nce, this is a boundary value equivalent of a sti problem. For this linear problem f (x y) = 0 0 : y The use of (8.2.6) reveals 1 ; =2 L( ) = L = ; h 1 0 ; 1 0 0 = ;1=h ;1=h 01 ; =2 2 1 = R( ) = R = ;=h2 ;=h2 i = 1 2 : : : N: =1 i i i i i i i Additionally, i i L(0 ) = L0 = 1 0] ) R( +1 = R N N +1 = 1 0]: Ascher et al. 1] solve a problem with = 50 by the midpoint and trapezoidal rules. The maximum error ke1 k = 0max jy1(x ) ; y1 j i 1 <i<N i i of the rst component of the solution is used as an accuracy measure. Results using uniform meshes are shown in Table 8.2.1. Although the convergence rate appears to be 6 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 8.2.2: Exact solution of Example 8.2.1 with = 50. The solid line is y1 and the dashed line is y2. quadratic, the accuracy is poor. With boundary layers near both ends (Figure 8.2.2), accuracy may be improved by using a nonuniform mesh that is concentrated near the endpoints. Ascher et al. 1] did this and their results are shown in Table 8.2.2. The results are somewhat better with convergence again appearing to be at a quadratic rate. Thus, a nonuniform mesh with the box scheme does not appear to reduce the convergence rate relative to that on a uniform mesh. This would not be the case with central di erencing. Results computed with the trapezoidal rule are approximately three times more accurate than those computed by the midpoint rule on a nonuniform mesh. 8.3 Higher-Order Methods and Deferred Corrections Let us once again consider the solution of the nonlinear BVP (8.2.1) by the trapezoidal rule (8.2.3). To begin, we'll recall that the local discretization error (De nition 6.3.1) of 7 N Midpoint Trapezoidal Rule Rule 10 0.47 0.44 20 0.20 0.19 40 0.057 0.056 80 0.01 0.012 Table 8.2.1: Errors in the solution of Example 8.2.1 using the midpoint and trapezoidal rules on uniform meshes. N Midpoint Trapezoidal Rule Rule 10 0.16 0.55 20 0.042 0.015 40 0.0096 0.0036 80 0.0024 0.00091 Table 8.2.2: Errors in the solution of Example 8.2.1 using the midpoint and trapezoidal rules on nonuniform meshes. the trapezoidal rule is = (y(x)) = y(x ) ; y(x 1) ; f (x y(x )) + f (x 1 y(x 1)) : (8.3.1) h 2 As usual, this may be put in a more explicit form by using Taylor's series expansions. i i i; i i i; i; i i Lemma 8.3.1. If f (x y) 2 C 2 K +2(a b) then X = h2 T (y(x 1 2 )) + O(h2 +2) K k i k where =1 (8.3.2a) K k i i; = i k T (y(x)) = ; 22 1(2k + 1)! f (2 ) (x y(x)): (8.3.2b) k k k; Proof. Expand the local discretization error in a Taylor's series about x 1 2 . i; = Thus, the local discretization error of the trapezoidal rule has a Taylor's series expansion in even powers of the mesh spacing. The leading term of this expansion is h2 = ; 12 f (x 1 2 y(x 1 2)): The result (8.3.2) does not depend on the mesh being uniform, but having some mesh regularity will provide bene ts. i i 00 i; = 8 i; = i N 1 0 x 0 x1 x2 1 = xN Figure 8.3.1: A mesh with quadratic grading. De nition 8.3.1. A sequence of meshes fa = x0 < x1 < : : : < x = bg with spacing h = x ; x 1, i = 1 2 : : : N , characterized by N ! 1 is c...
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