If we know that all of the eigenvalues of a sti

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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) step-size 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 non-sti 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...
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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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