Geometric Integrators with Application to Hamiltonian Systems
Geometric numerical integration is a relatively new area of numerical analysis. The aim is to preserve the geometric properties of the flow of a differential equation such as symplecticity or reversibility. A conventional numerical integrator approximates the flow of the continuous-time equations using only the information about the vector field, ignoring the physical laws and the properties of the original trajectory. In this way, small inaccuracies accumulated over long periods of time will significantly diminish the operational lifespan of such discrete solutions. Geometric integrators, on the other hand, are built in a way that preserve the structure of continuous dynamics, so maintaining the qualitative behavior of the exact flow even for long-time integration. The aim of this thesis is to design efficient geometric integrators for Hamiltonian systems and to illustrate their effectiveness. These methods are implicit for general (non-separable) Hamiltonian systems making them difficult to implement. However, we show that explicit integrators are possible in some cases. Both geometric and non-geometric integration methods are applied to several problems, then we do a comparison between these methods, in order to determine which of those quantities are preserved better by these methods. In particular, we develop explicit integrators for a special case of the restricted 3-body problem known as Hill’s problem.