Geometric Integrators for Non-separable Hamiltonian Systems

In this thesis, we consider non-separable Hamiltonian systems, and we develop an integrator that combines Pihajoki’s expanded approach to phase space with the symmetric projection technique. Through this, we construct a semiexplicit numerical integrator, meaning that the primary time evolution step is explicit but the symmetric projection step is implicit. The symmetric projection fixes the major disadvantage of the extended phase space technique by binding possibly divergent copies of solutions. In addition, our semiexplicit approach gives the first extended phase space integrator that is symplectic in the original phase space. This is in contrast to those explicit extended phase space integrators of Pihajoki and Tao, which are symplectic only in the extended phase space. Our integrator tends to preserve invariants better than Tao’s. Moreover, for some higher-order implementations and higher-dimensional problems, ours is faster than Tao’s explicit method despite being partially implicit.