SEMIEXPLICIT SYMPLECTIC INTEGRATORS FOR NON-SEPARABLE HAMILTONIAN SYSTEMS

We construct a symplectic integrator for non-separable Hamilton-ian systems combining an extended phase space approach of Pihajoki and the symmetric projection method. The resulting method is semiexplicit in the sense that the main time evolution step is explicit whereas the symmet-ric projection step is implicit. The symmetric projection binds potentially diverging copies of solutions, thereby remedying the main drawback of the ex-tended phase space approach. Moreover, our semiexplicit method is symplec-tic in the original phase space. This is in contrast to existing extended phase space integrators, which are symplectic only in the extended phase space. We demonstrate that our method exhibits an excellent long-time preservation of invariants, and also that it tends to be as fast as and can be faster than Tao's explicit modified extended phase space integrator particularly for small enough time steps and with higher-order implementations and for higher-dimensional problems.

Automated summary

In this paper, we propose a symplectic integrator for non-separable Hamilton-ian systems, which combines the extended phase space approach of Pihajoki and the symmetric projection method. Our method is semiexplicit, with an explicit main time evolution step and an implicit symmetric projection step, and it preserves symplecticity in the original phase space.