abs.tex

\begin{abstract}
    Geometric methods for integrating dynamical systems aim to preserve behaviour from the system in the numerical solution.
    General-purpose integration schemes are introduced in order to understand the behaviour of numerical solutions to ODEs and build the foundation for geometric integration.
    Symplectic integrators develop from the structure of Hamiltonian systems.
    These methods are generally implicit but can be explicit if the Hamiltonian is separable.
    Examples show the utility of symplectic methods in physical scientific applications.
    Backward error analysis shows that the symplectic method exactly solves a problem whose modified Hamiltonian has closeness to the original depending on the order of the method used.
    Positivity preserving methods in our analysis are formulated for problems involving a graph-Laplacian matrix.
    Methods which are of current interest are second order, which is shown, and employ the matrix exponential.
    Two adjustments to a positivity preserving method are proposed,
    where if an approximation of the matrix exponential will unconditionally preserve positivity then it can be employed in order to reduce cost.
\end{abstract}