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CRGreathouse (talk | contribs) m Unlinked: Explicit, Implicit using Dab solver |
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& \qquad {} = h \bigl( b_s f(t_{n+s},y_{n+s}) + b_{s-1} f(t_{n+s-1},y_{n+s-1}) + \cdots + b_0 f(t_n,y_n) \bigr),
\end{align} </math>
a good approximation of the differential equation <math> y' = f(t,y) </math>? More precisely, a multistep method is ''consistent'' if the local error goes to zero as the step size ''h'' goes to zero, where the ''local error'' is defined to be the difference between the result <math>y_{n+s}</math> of the method, assuming that all the previous values <math>y_{n+s-1}, \ldots, y_n</math> are exact, and the exact solution of the equation at time <math>t_{n+s}</math>, divided by h. A computation using [[Taylor series]] shows out that a linear multistep method is consistent if and only if
:<math> \sum_{k=0}^{s-1} a_k = -1 \quad\text{and}\quad \sum_{k=0}^s b_k = s + \sum_{k=0}^{s-1} ka_k. </math>
All the methods mentioned above are consistent {{harv|Hairer|Nørsett|Wanner|1993|loc=§III.2}}.
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