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Unformatted text preview: ough Runge-Kutta formulas are tedious to derive, we can make a few general
observations. An order one formula must be exact when the solution of the ODE is a
linear polynomial. Were this not true, it wouldn't annihilate the constant and linear
10 terms in a Taylor's series expansion of the exact ODE solution and, hence, could not
have the requisite O(h2) local error to be rst-order accurate. Thus, the Runge-Kutta
method should produce exact solutions of the di erential equations y = 0 and y = 1.
The constant-solution condition is satis ed identically by construction of the RungeKutta formulas. Using (3.2.3a), the latter (linear-solution) condition with y(t) = t and
f (t y) = 1 implies
tn = tn 1 + h bi
0 ; i=1 or
i=1 0 bi = 1: (3.2.5a) If we also require the intermediate solutions Yi to be rst order, then the use of (3.2.3b)
with Yi = tn 1 + cih gives
ci = aij
i = 1 2 : : : s:
; j =1 This condition does not have to be satis ed for low-order Runge-Kutta methods 16]
however, its satisfaction simpli es the task of obtaining order conditions for higher...
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- Spring '14