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Unformatted text preview: and consistency
implying convergence. However, in contrast, the local and global discretization errors
converge at the same rates. 8.2 The Box Scheme for First-Order Systems
We'll extend nite di erence methods to nonlinear rst-order BVPs of the form y = f (x y) a<x<b (8.2.1a) g (y(a)) = g (y(b)) = 0: (8.2.1b) 0 L R As in Chapter 7, y and f are m-vectors, g is an l-vector, and g is an r-vector with
l + r = m.
We will discretize (8.2.1) by a variant of the trapezoidal rule called the \box scheme"
that was proposed by Keller 6] and developed by Lentini and Pereyra 7]. Towards this
end, introduce a partition of (a b) into N subintervals as shown in Figure 8.2.1. Let
L 3 R y yi-1
yi y(x) hi
a = x0 x1 x i-1 xi x N= b Figure 8.2.1: Mesh nomenclature for the box scheme. h =x ;x
i i (8.2.2) 1 i; and approximate (8.2.1a) using the trapezoidal rule
(y) = y ; y 1 ; f (x y ) + f (x 1 y 1) = 0
y = 6 ..1 7 :
i i; i i i; i; i i i = 1 2 ::: N (8.2.3a) (8.2.3b) y N The boundary conditions (8.2.1b) complete the speci cation of the discrete problem as g (y0 ) = g (y ) = 0:
L R (8.2.3c) N The BVP has been approximated by a nonlinear algebraic system (8.2.3) of dimension
mN for the N vector of unknowns y , i = 0 1 : : : N .
The midpoint rule can be used in place of the trapezoidal rule to obtain
(y) = y ; y 1 ; f (x 1 2 y + y 1 ) = 0:
The midpoint and trapezoidal rules have similar, but not identical, accuracy and stability
i i i i; i; = i i 4 i; Focusing on the trapezoidal rule, we'll linearize (8.2.3) by Newton's method. For
(8.2.3a) we have
@ (y( ) ) y( ) = ; (y( ) )
i = 1 2 ::: N
and for (8.2.3b) we have
i i (
@ g (y0 )) y( ) = ;g (y( ) )
@ y0 (8.2.5b) @ g (y( )) y( ) = ;g (y( ))
@y (8.2.5c) y( +1) = y( ) + y( ) : (8.2.5d) L L R N R N N N with Using (8.2.3a), the two nonzero contributions to the Jacobian in (8.2.5a) are @ (y( ) ) := L( ) = ; I ; fy (x 1 y( )1)
i; i (8.2.6a) i; i i; i and @ (y( ) ) := R( ) = I ; fy (x y( ))
where I is the m m identity matrix.
For consistency and simplicity, we'll also write the Jacobians in (8.2.5b,c) as
i i i i i i L( ) :=
@ gL(y0 ) )
@ y0 (8.2.6c) (
@ gR (yN ) )
Collecting (8.2.5) and (8.2.6), we nd the linear Newton system as
R(N+1 := 2
( ) R( ) 7 4
5 L(0 )
L(1 ) R(1 ) N )
g (y0 ) )
6 (y( ) )
7 = ;6 1 .
4 (y( ) )
g (y ( ) )
y1 ) N N N 5 L R N (8.2.6d)
5 (8.2.7) We'll discuss an O(N ) algorithm for solving this block bidiagonal system in Section 8.5.
For the moment, we'll consider an application.
Example 8.2.1 ( 1], Section 5.1). Consider the second-order linear BVP y ; 2y = 2 cos2 x + 2 2 cos 2 x 0<x<1 00 y(0) = y(1) = 0: We'll write this as a rst-order system by letting y1 = y and y2 = y= to get y1
y2 0 y1 +
y2 = 00 cos2 02
x + 2 cos 2 x : The exact solution of this problem is
" ; (1;x) + ; x
; cos2 x :
y(x) = ; (1;x) ; x + sin 2 x
e e e e ;e e This problem is similar to one that was addressed in Section 7.1. We recall that when
1 there are boundary layers where the solution varies rapidly near both x = 0 and
x = 1. The solution varies sinusoidally in the interior (Figure 8.2.2). He...
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.
- Spring '14