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% \it Maths 3018/6111 - Numerical methods \hfill}
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\textbf{\Large Maths 3018/6111 - Numerical Methods \\*[8mm]
Worksheet 5}\\*[.8cm]
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\section*{Theory}

\begin{enumerate}
\item Explain when multistep methods such as Adams-Bashforth are
  useful and when multistage methods such as RK methods are better.
\item Compute the coefficients of the AB3 algorithm.
\item Explain the meaning of stability, consistency and convergence
  when applied to numerical methods for IVPs. State the theorem
  connecting these.
\item Using the stability polynomial and your results above, check the
  order of accuracy and the stability of the 3 step Adams-Bashforth
  method. 
\end{enumerate}

\section*{Coding}

\begin{enumerate}
\item Apply the 2-step Adams-Bashforth method to the ODE from
  Worksheet 4, 
  \begin{equation*}
    y' + 2 y = 2 - e^{-4 x}, \quad y(0) = 1.
  \end{equation*}
  Use the Euler or Euler predictor-corrector method to start. Again,
  find the value of $y(1)$ (analytic answer is $1 -
  (e^{-2}-e^{-4})/2$) and see how your method converges with
  resolution.
\item Apply the 2-step implicit Adams-Moulton method to the above ODE,
  using the 2-step Adams-Bashforth method as a predictor. Use the
  Euler or Euler predictor-corrector method to start. See how your
  method converges with resolution.

\end{enumerate}

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