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Unformatted text preview: i.e.,
i N i 1 i 1 i w(x y(x)) = w i x2 x i; 1 x)
i i = 1 2 : : : N: This situation is shown in Figure 8.4.3. With a piecewise constant monitor function, I (x)
18 would be the piecewise linear function I (x) = I (x0 1) + w (x ; x0 1)
j; j x 2 x0 1 x0 ) j; j; j as shown in Figure 8.4.4. In this case, (8.4.5) can be solved exactly. For example, suppose
I(x)
I(x N) w2 wN w1
x
xo
1 xo
0 xo
2 xo
N Figure 8.4.3: Piecewise constant monitor function.
I(x)
I(x N) I(x 1 ) x
xo
0
x0 xo
1
x1 xo
N
xN xo
2
x2 Figure 8.4.4: Piecewise linear function I (x) based on a piecewise constant monitor function. I (x0 1) i < I (x )
j j; 19 then i ; I (x0 1)
:
w
A piecewise linear interpolation of w , i = 1 2 : : : N , is also possible, e.g.,
x0 ; x
x ; x0 1
w(x y(x)) = w 1 x0 ; x0 + w x0 ; x0 :
1
1
In this case, I (x) is a piecewise quadratic function of x and the roots of (8.4.5) can still
be determined exactly 2].
Remark 1. You may think of using an iterative process where the equidistributing
mesh obtained as described above is used as an input mesh to calculate a \better"
equidistributing mesh, etc. Coyle et al. 2] show that this procedure is not likely to
converge.
The dimensions of the input and output meshes need not be the same. One could,
for example, calculate an estimate of the number of points needed in the output mesh to
reduce the global error to a desired tolerance. Let us simplify our previous notation and
suppose that an error indicator of the form
x = x0 1 +
i j; j; j i j j; j; j j j; j j; e = d (h0) s j j j is an estimate of the global error (in, e.g., the maximum norm) on the subinterval
x0 1 x0 ). The constant d includes the dependence of the error on derivatives of the
solution. If e > TOL where TOL is the prescribed tolerance, we could calculate a new
mesh spacing
h0
h =N
j; j j i j j j and determine the re nement factor N such that
j dh T OL: s j Thus, N j j d
h0( TOL )1 :
j =s j The N points could be added to the subinterval x0 1 x0 ) or used to determine the
dimension of an equidistributing mesh. In a similar manner, points can be removed from
subintervals that have small errors.
j j; 20 j 8.5 Solving Block Bidiagonal Linear Systems
In Section 8.2, we saw that the solution of a vector system of rstorder di erential equations by the trapezoidal rule required the solution of a block bidiagonal linear algebraic
system of the form Ay = b
where 2
L0
6 L1 R 1
6
... ...
A=6
6
6
4
L N R
R +1
N 3
7
7
7
7
7
5 2
6
y=6
6
4 (8.5.1a) y0
y1
... y N N 3
7
7
7
5 2
3
b0
6 b1 7
6
6 . 7:
b = 6 .. 7
7
6
4b 7
5 b (8.5.1b) N N +1 The matrices L and R , i = 1 2 : : : N , have dimension m m, but the matrix L0
has dimension l m and R +1 has dimension r m where l + r = m. Likewise, b0
is an lvector, b , i = 1 2 : : : N , are mvectors, and b +1 is an rvector. Each y ,
i = 0 1 : : : N , is an mvector. The zerononzero structure of the matrix A is shown in
Figure 8.5.1 for a case when l = r = 1 and m = 2.
Systems of this form also arise in conjunction with multiple shooting procedures
(Section 7.2) and with collocation methods (Section9.1).
Keller 6] embedded the block bidiagonal matrix A in a...
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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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