Linear multistep method: Difference between revisions

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If the method is consistent, then the next question is how well the difference equation defining the numerical method approximates the differential equation. A multistep method is said to have ''order'' ''p'' if the local error is of order <math>O(h^{p+1})</math> as ''h'' goes to zero. This is equivalent to the following condition on the coefficients of the methods:

:<math> \sum_{k=0}^{s-1} a_k = -1 \quad\text{and}\quad q k^{q-1} \sum_{k=0}^s k^{q-1} b_k = s^q + \sum_{k=0}^{s-1} k^q a_k \text{ for } q=1,\ldots,p. </math>

The ''s''-step Adams–Bashforth method has order ''s'', while the ''s''-step Adams–Moulton method has order <math>s+1</math> {{harv|Hairer|Nørsett|Wanner|1993|loc=§III.2}}.