Chapter 2

Derive explicitly the three-step and four-step Adams-Moulton methods and the three-step Adams-Bashforth method.

By solving a three-term recurrence relation, calculate analytically the sequence of values \(y_{2}, y_{3}, y_{4}, \ldots\) that is generated by the midpoint rule $$ \boldsymbol{y}_{n+2}=\boldsymbol{y}_{n}+2 h \boldsymbol{f}\left(t_{n+1}, \boldsymbol{y}_{n+1}\right) $$ when it is applied to the differential equation \(y^{\prime}=-y\). Starting from the values \(y_{0}=1, y_{1}=1-h\), show that the sequence diverges as \(n \rightarrow \infty .\) Recall, however, from Theorem \(2.1\) that the root condition, in tandem with order \(p \geq\) 1 and suitable starting conditions, imply convergence to the true solution in a finite interval as \(h \rightarrow 0+\). Prove that this implementation of the midpoint rule is consistent with the above theorem. [Hint: Express the roots of the characteristic polynomial of the recurrence relation as \(\exp \left\\{\pm \sinh ^{-1} h\right\\} .\) ]