Unformatted text preview: lt; Re(h ) <
DR Im(h )j < DI g.
The regions R1 and R2 are shown in Figure 5.6.7. Those eigenvalues that represent
rapidly decaying terms in the transient solution of the ODE may be solved with step
43 Im(h λ
) DI R1 R2
Re(h λ
) DL
DR D I Figure 5.6.7: Schematic of a region of sti stability.
sizes h such that h 2 R1 . These components would be approximated stably but not
accurately. Eigenvalues of the system representing slowly varying parts of the solution
are smaller and their product with h would place them in R2 . Values of h that are
outside of R1 R2 should not be used since either the growth of the solution is too
fast (when Re(h ) > 0) to be approximated accurately or the solution is too oscillatory
(when jIm(h )j > DI ) to follow accurately. In both cases, h should be reduced so that
h 2 R2 .
Example 5.6.7. From Figure 5.6.4, we see that the backward di erence formulas of
orders one through six are sti y stable. They are also A( )stable, but decreases with
increasing order.
Example 5.6.8. From (5.5.5), we recall that the backward di erence formulas have
the form
k
X
i=0 i yn;i = 0 fn : Applying this method to the test equation (5.6.4) yields
k
X
i=0 i yn;i ; h 44 0 yn = 0: Assuming a solution of the form n gives
( );h ( )=0 with
( )= k
X
i=0 i k;i ( )= 0 k: As Re(h ) ! ;1, the roots satisfy
( ) = 0:
The roots of ( ) = 0 are i = 0, i = 1 2 : : : k. So all solutions of the backward
di erence formulas decay as Re(h ) ! ;1. 5.7 Implementation: Error and step size control
There are several implementation issues to discuss, including (i) starting values, (ii)
iteration for implicit methods, (iii) error estimation, (iv) stepsize variation, and (iv)
method order variation. We'll begin with the solution of implicit di erence equations,
which are essential for sti systems, but also preferred for nonsti ODEs because of their
smaller error coe cients and larger regions of absolute stability. Consider an implicit
LMM of the form (5.1.2) and write it as yn + k
X
i=1 i...
View
Full
Document
This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.
 Spring '14
 JosephE.Flaherty
 The Land

Click to edit the document details