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

TL;DR

This paper introduces a general operator splitting method for solving autonomous and non-autonomous equations, enabling high-order algorithms up to 100th order, with applications demonstrated on quantum and Coulomb problems.

Contribution

It develops a unified operator-based framework for multi-product splitting, deriving high-order integrators including odd orders, and demonstrates their effectiveness on complex physical systems.

Findings

High-order algorithms up to 100th order are successfully tested.

The method shows uniform convergence for singular Coulomb problems.

It provides a structure non-preserving approach that avoids poles in solutions.

Abstract

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.