  Symplectic integrators that preserve the geometric structure of Hamiltonian
flows and do not exhibit secular growth in energy errors are suitable for the
long-term integration of N-body Hamiltonian systems in the solar system.
However, the construction of explicit symplectic integrators is frequently
difficult in general relativity because all variables are inseparable.
Moreover, even if two analytically integrable splitting parts exist in a
relativistic Hamiltonian, all analytical solutions are not explicit functions
of proper time. Naturally, implicit symplectic integrators, such as the
midpoint rule, are applicable to this case. In general, these integrators are
numerically more expensive to solve than same-order explicit symplectic
algorithms. To address this issue, we split the Hamiltonian of Schwarzschild
spacetime geometry into four integrable parts with analytical solutions as
explicit functions of proper time. In this manner, second- and fourth-order
explicit symplectic integrators can be easily made available. The new
algorithms are also useful for modeling the chaotic motion of charged particles
around a black hole with an external magnetic field. They demonstrate excellent
long-term performance in maintaining bounded Hamiltonian errors and saving
computational cost when appropriate proper time steps are adopted.
