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Unformatted text preview: a methods may also have expansions of this form.
c t j ::: p p<q
yt 1 The idea of extrapolation is to combine solutions ( j ) obtained with di erent
step sizes j , = 0 1 , so that successive terms of the error expansion (4.1.2) are
eliminated thus, resulting in a higherorder approximation. The process is commonly
called Richardson's extrapolation. Let us begin with a simple example.
Example 4.1.1. If solutions are smooth, the global error of either the forward or
backward Euler methods have series expansions of the form
zth h j ::: ( ) = ( )+ zth yt + c1 h 2 c2 h + ::: : Obtain two solutions at the same time using step sizes of
t ( z t h0 )= ( )+
yt c1 h0 + 2 c2 h0 + h0 and 2, i.e., h0 = ::: + 2 ( 0 )2 +
2
2
Subtracting these two equations, we eliminate ( ) and obtain
10
= ( 0 ) ; ( 0 2) ; 342 2 +
0
2
The leading term of this expression furnishes a discretization error estimate of either
solution. Substituting the above expression into either of the two global error expansions
yields
2 ( 0 2) ; ( 0 ) = ( ) ; 22 2 +
0
( z t h0 = 2) = ( ) +
yt c1 h0 c h ::: : yt ch zth zth= zth= zth yt c c h ::: : h ::: : Thus, 2 ( 0 2) ; ( 0 ) provides a higherorder ( ( 2)) approximation of ( ) than
either ( 0 ) or ( 0 2).
The same result can be obtained by approximating ( ) by a linear polyno...
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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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