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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 HigherOrder 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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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.
 Spring '14
 JosephE.Flaherty

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