Linear multistep method: Difference between revisions

Revision as of 18:32, 15 March 2016

Revision as of 09:45, 3 November 2016

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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 truncation error]] goes to zero faster than the step size ''h'' as ''h'' goes to zero, where the ''local truncation 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>. 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}}.