Unformatted text preview: = ; 38 2 + qi;1=2
h 2p i
q
= 34 2 ; 3;1=2 + i;1=2
h
h
6
0 =2
3 i i = 1 2 ::: N
=2
3 (6.4.14c) i = 1 2 : : : N: (6.4.14d) i = 1 2 ::: N 2pi
q
= 34 2 + 3;1=2 + i;1=2
h
h
6 (6.4.14b) N +1 As a simple numerical example, let's suppose p = 0, q = ;1, r = 0, A = 0, and
B = 1. This is the problem of Example 6.3.2 which has the exact solution y(x) = sinh x :
sinh 1
Solution and pointwise (at xi , i = 0 1 : : : N ) error data are presented in Table 6.4.1
for N = 10. The maximum pointwise errors for collocation solutions computed with
38 i xi
Y (xi)
y(xi) ; Y (xi)
0 0.0 0.0000000 0.0000000E+00
1 0.1 0.0852282 0.5491078E05
2 0.2 0.1713098 0.1069903E04
3 0.3 0.2591066 0.1528859E04
4 0.4 0.3494977 0.1892447E04
5 0.5 0.4433881 0.2130866E04
6 0.6 0.5417180 0.2211332E04
7 0.7 0.6454719 0.2080202E04
8 0.8 0.7556885 0.1698732E04
9 0.9 0.8734714 0.1043081E04
10 1.0 1.0000000 0.2384186E06
Table 6.4.1: Solution and pointwise errors for Example 6.4.1 using collocation with piecewise quadratic splines at the center of each of ten subintervals.
N
kY ; yk1
N 2 kY ; yk1
5 0:8890263 10;4 0.002223
10 0:2214184 10;4 0.002214
20 0:5537689 10;5 0.002215
Table 6.4.2: Maximum pointwise errors for the solution of Example 6.4.1 using collocation
with piecewise quadratic splines at the center of each of N subintervals.
N = 5 10, and 20 subintervals are presented in Table 6.4.2. Solutions appear to be
converging as O(h2). Pointwise errors are about half of those found using central nite
di erences (Table 6.3.1).
Example 6.4.2. In the previous example, it seemed natural to place the single collocation point at the center of each subinterval. Were we to used piecewise cubic Hermite
approximations, however, we would have two collocation points per subinterval. Placing
them at the ends of each subinterval is one possibility however, our work with implicit
RungeKutta methods (Section 3.3) would suggest that the GaussLegendre points give
a higher rate of convergence. DeBoor and Swartz 2] showed that this is the case and we
will repeat this analysis in Chapter 9 however, for the moment, let us assume it so and
consider the collocation solution of ( 1], Chapter 5)
0
8
0<x<1
y00 + y = ( 8 ; x2 )2
x
y0(0) = y(1) = 0
39 N jjY ; yjj1
2 0:20 10;3
5 0:64 10;5
10 0:46 10;6
20 0:33 10;7
40 0:23 10;8
80 0:16 10;9
Table 6.4.3: Maximum pointwise errors for the solution of Example 6.4.2 using collocation
with piecewise cubic Hermite polynomials at two GaussLegendre points per subinterval.
which has the exact solution 7
y(x) = 2 ln( 8 ; x2 ): The solution is smooth on 0 x 1, but the coe cient p(x) = 1=x is unbounded at
x = 0. This would lead to problems with nite di erence or shooting methods, but not
with collocation methods that collocate at points other than subinterval ends. Ascher
et al. 1] solve this problem by a variety of techniques. We'll report their results using
piecewise cubic Hermite polynomials with collocation at the two GaussLegendre points
1
1
h
h
i 2 = (xi;1 + xi ) + p ]
i 1 = (xi;1 + xi ) ; p ]
2
2
3
3
per subinterval. The maximum pointwise errors are presented in Table 6.4.3. A simple
calculation veri es that the solution is converging as O(N ;4). Problems 1. Construct the basis for a cubic spline approximation that is of class C 2 and has
support on xi;2 xi+2 ] by integrating the quadratic splines 2 (x) and 2+1(x) of
i
i
(6.4.6) and imposing appropriate normalization and compact support conditions.
For simplicity, assume that the mesh is uniform with spacing h.
2. Using (6.4.7c) show that 2
Y (xi ) = 3 (ci + ci+1) on a uniform mesh of spacing h. Use this to rewrite the quadraticspline collocation
equations (6.4.13) in terms of Y (xi), i = 1 2 : : : N , instead of ci, i = 0 1 : : : N +
1. Compare the results with the nite di erence equations (6.3.14).
40 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 and B. Swartz. Collocation at gaussian points. SIAM J. Numer. Anal.,
10:582{687, 1973.
3] C. deBoor. A Practical Guide to Splices. SpringerVerlag, Berlin, 1978. 41...
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 Spring '14
 JosephE.Flaherty
 Numerical Analysis, Boundary value problem, di erences, di erence

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