plan.tex

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\begin{document}

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\title{Report Plan}
\author{Joel Biffin}
\maketitle

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\section{Background}

\begin{itemize}
  \item{Definition of IVPs (standard 1st order form).}
  \item{Where they come from (modelling natural phenomena).}
  \item{Reducing nth order IVPs to standard (vector) form.}
  \item{How evolutionary problems modelled by PDEs can be reduced to discretised IVPs.}
\end{itemize}

\section{Numerical Methods}

\begin{itemize}
  \item{One-step methods, explicit vs implicit.}
  \item{Local truncation error and the order of a method.}
  \item{Stability: region of stability, A-stability and L-stability.}
  \item{Runge-Kutta methods.}
  \item{Linear multi-step methods: Adams’, BDF, Nystrom.}
  \item{Predictor corrector methods and pairings.}
  \item{Variable time-mesh.}
\end{itemize}

\section{Software Design}

\begin{itemize}
  \item{Choice of python and numpy.}
  \item{Why choose to use object-oriented framework (future considerations).}
  \item{Briefly discuss design using UML.}
  \item{Difficulties that this presented in development.}
  \item{Graphing results (very brief).}
\end{itemize}

\section{Analysis of Numerical Methods}

\begin{itemize}
  \item Use scalar examples to demonstrate stability, stiffness and natural characteristics of each method.
     \begin{itemize}[label=$\star$]
        \item Non-stiff example, showing Forward Euler method vs Adams-Bashforth method with same step-size – graph Local truncation errors too.
        \item Stiff example, showing Forward Euler vs Backward Euler with multiple step-sizes (emphasising region of stability).
        \item Stiff example, similarly for Adams-Bashforth vs Adams-Moulton.
        \item Stiff example, show Adams-Moulton vs BDF2 – graph local truncation errors.
     \end{itemize}
  \item Showing comparison of predictor-corrector pairings with fixed mesh – graph local truncation errors.
    \begin{itemize}[label=$\star$]
      \item Plot graph of step-sizes against local-truncation errors.
    \end{itemize}
  \item Showing comparison of predictor-corrector pairings with variable mesh.
    \begin{itemize}[label=$\star$]
      \item Demonstrate how milne’s device is used with set tol and its impact on accuracy and stability of methods.
    \end{itemize}
\end{itemize}

\section{Case Studies}

\begin{itemize}
  \item Single pendulum (no damping-simple harmonic motion).
    \begin{itemize}[label=$\star$]
      \item Demonstrate natural damping feature of fixed step Forward-Backward Euler predictor-corrector, vs Adams-Bashforth-Moulton-2 predictor corrector (no damping).
    \end{itemize}
  \item Damped single pendulum.
  \item Damped forced pendulum.
  \item Double pendulum (chaotic motion).
\end{itemize} 


\section{Future Development}

\begin{itemize}
  \item Choice of step-size update function and local truncation error estimate in predictor-corrector (limitations to Milne’s device).
  \item Refactoring code to use Generalised LMMs and k-step BDF methods with variable coefficients so that the order of methods can be updated on-the-fly dependent on local truncation error estimates.
  \item Refactoring code to not store the entirety of results in memory – automated garbage collection or saving to disc. 
\end{itemize}


\end{document}