Math 6620 • Analysis of Numerical Methods II

(20 pts) For the ODE \(u'(t) = f(t,u)\), consider the scheme,
\[
u_{n+1}+\alpha_1u_n = k\beta_0 f_{n+1}+k\beta_1 f_n,
\]
where \(u_n \approx u(t_n)\), \(f_n = f(t_n,u_n)\), \(k = t_{n+1}-t_n\), and \(u' = f(t,u)\).
\begin{enumerate}

\item
(10 pts) Identify the coefficients \(\alpha, \beta\) that yield a scheme of optimal \(k\)-order of accuracy,
and identify this order of accuracy.


\item
(10 pts) Determine whether or not this scheme is 0-stable and/or A-stable.

\end{enumerate}