Chapter 2

Determine the order of the three-step method $$ \begin{gathered} \boldsymbol{y}_{n+3}-\boldsymbol{y}_{n}=h\left[\frac{3}{8} \boldsymbol{f}\left(t_{n+3}, \boldsymbol{y}_{n+3}\right)+\frac{9}{8} \boldsymbol{f}\left(t_{n+2}, \boldsymbol{y}_{n+2}\right)+\frac{9}{8} \boldsymbol{f}\left(t_{n+1}, \boldsymbol{y}_{n+1}\right)\right. \\ \left.+\frac{3}{8} \boldsymbol{f}\left(t_{n}, \boldsymbol{y}_{n}\right)\right] \end{gathered} $$ the three-eighths scheme. Is it convergent?

By solving a three-term recurrence relation, calculate analytically the sequence of values \(y_{2}, y_{3}, \ldots\) 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 \(\left.\exp \left(\pm \sinh ^{-1} h\right) .\right)\)

Find the explicit form of the BDF for \(s=4\).