Multi-product operator splitting as a general method of solving autonomous and non-autonomous equations
Abstract
Prior to the recent development of symplectic integrators, the time-stepping operator h(A+B) was routinely decomposed into a sum of products of h A and 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 even and odd order Runge-Kutta and Nyström 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 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 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.