Math 6620 • Analysis of Numerical Methods II

(20 pts) For the ODE \(u'(t) = f(t,u)\), consider the time-integration scheme
\[
u_{n+1} = u_n + \frac{k}{4}f(t_n,u_n)
+ \frac{3k}{4}f\left(t_n+\frac{2}{3}k,\ u_n+\frac{2}{3}kf(t_n,u_n)\right).
\]
\begin{enumerate}

\item
(5 pts) Identify the Butcher tableau for this scheme.


\item
(8 pts) Compute the order of consistency of this scheme.


\item
(7 pts) Compute the stability/amplification factor of the scheme.

\end{enumerate}