Math 6620 • Analysis of Numerical Methods II

For the initial value problem \(u'(t) = f(u(t),t)\), \(u(0) = \eta\), where \(f(u,t)\) is continuous with respect
to \(t\) and Lipschitz continuous with respect to \(u\), consider the scheme
\[
\frac{U^{n+2}-\frac{4}{3}U^{n+1}+\frac{1}{3}U^n}{k}
= \frac{2}{3}f(U^{n+2},t_{n+2}),
\]
where \(t_n = nk\), and \(U^n\) is supposed to approximate \(u(t_n)\).
\begin{enumerate}

\item
Analyze the consistency, zero-stability, and convergence of this scheme.


\item
If you apply this scheme to the initial value problem

\[
u'(t) = -10^{12}(u(t)-\cos(t))-\sin(t),\qquad u(0) = 2,
\]


what issues should you consider in choosing the timestep \(k\)? Are there time intervals in which \(k\)
must be very small to get a reasonable solution and others in which it does not? Explain your
answers.

\end{enumerate}