# Implicit runge kutta methods may also have expansions

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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 higher-order 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 higher-order ( ( 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.

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