Multi-product operator splitting as a general method of solving autonomous and non-autonomous equations

Prior to the recent development of symplectic integrators, the time-stepping operator $\e^{h(A+B)}$ was routinely decomposed into a sum of products of $\e^{h A}$ and $\e^{hB}$ in the study of hyperbolic partial differential equations. In the context of solving Hamiltonian dynamics, we show that such a decomposition give rises to both {\it even} and {\it odd} order Runge-Kutta and Nystr\"om integrators. By use of Suzuki's forward-time derivative operator to enforce the time-ordered exponential, we show that the same decomposition can be used to solve non-autonomous equations. In particular, odd order algorithms are derived on the basis of a highly non-trivial {\it time-asymmetric} kernel. Such an operator approach provides a general and unified basis for understanding structure non-preserving algorithms and is especially useful in deriving very high-order algorithms via {\it analytical} extrapolations. In this work, algorithms up to the 100th order are tested by integrating the ground state wave function of the hydrogen atom. For such a singular Coulomb problem, the multi-product expansion showed uniform convergence and is free of poles usually associated with structure-preserving methods. Other examples are also discussed.