On the local error and the local truncation error of linear multistep methods (Q753437)

For linear multistep methods two different measures for the local accuracy are compared: the local truncation error whose leading term is that of \(y_{n+k}-y(x_{n+k})\) where \(y_{n+k}\) is the numerical result of the method when exact starting values \(y_ j=y(x_ j)\) for \(j=n,...,n+k-1\) are used. The local error is the difference \(y_{n+k}- y(x_{n+k})\) where \(y_{n+k}\) is obtained from the previously computed values \(y_ n,...,y_{n+k-1}\) and y(x) is the solution passing through \(y(x_{n+k-1})=y_{n+k-1}.\) The conclusion of the author is that the leading terms of the local truncation error and of the local error are in general different, but for a subclass, including the methods of Adams type, they are identical.