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Unformatted text preview: ries and substituting
the result into the above expression yields
; ; y(tn) = y(tn 1) + hb1 f + c1 hft + ha11 ffy + O(h2)]:
; Comparing the terms of the above series with the Taylor's series
h2 (f + ff ) + O(h3)
y(tn) = y(tn 1) + hf + 2 t
of the exact solution yields
b1 = 1
a11 = c1 = 2 :
Substituting these coe cients into (3.3.1), we nd the method to be an implicit midpoint
; yn = yn 1 + hf (tn 1 + h=2 Y1)
; ; Y1 = yn 1 + h f (tn 1 + h=2 Y1):
The tableau for this method is
; ; 19 (3.3.2a)
2 1 The formula has similarities to the midpoint rule predictor-corrector (3.1.11) however,
there are important di erences. Here, the backward Euler method (rather than the
forward Euler method) may be regarded as furnishing a predictor (3.3.2b) with the
midpoint rule providing the corrector (3.3.2a). However, formulas (3.3.2a) and (3.3.2b)
are coupled and must be solved simultaneously rather than sequentially.
Example 3.3.2. The two-stage method having maximal order four presented in th...
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- Spring '14