In this paper solutions to dynamical systems obeying Hamiltonian mechanics are obtained using algorithms from numerical integration. Symplectic integrators are preferred and applied to test problems such as a simple pendulum and a spring pendulum. Ascertaining the most computationally efficient route to solution has traditionally been a trade off between a employing higher order integrators and lowering the time step at a set order. It is proposed that a new set of time reversible integrators, which solve a modified Hamiltonian instead of the exact, may perform better than standard symplectic integrators of an equivalent order. Our investigation of the performance of the new schemes against established integrators on the test problems finds no evidence to support this claim. The study moves on to analyse the effect of discrete jumps in energy of the numerical solution, which exist for certain trajectories of the spring pendulum, on the performance of the integrators. Finally we study numerical solutions to a Lennard-Jones fluid, where energy jumps are prevalent and the system provides a much sterner test for the integrators than the two pendulum systems.
BSc in Mathematics and Physics, Faculty of Science, University of Warwick, July 2007
Joint project with Nichola Roberts
Supervised by Professor Robin Ball