Symplectic analytically integrable decomposition algorithms: classification, derivation, and application to molecular dynamics, quantum and celestial mechanics simulations Original Research Article

A complete classification of all explicit self-adjoint decomposition algorithms with up to 11 stages is given and the derivation process used is described. As a result, we have found 37 (6 non-gradient plus 31 force-gradient) new schemes of orders 2 to 6 in addition to 8 (4 non- and 4 force-gradient) integrators known earlier. It is shown that the derivation process proposed can be extended, in principle, to arbitrarily higher stage numbers without loss of generality. In practice, due to the restricted capabilities of modern computers, the maximal number of stages, which can be handled within the direct decomposition approach, is limited to 23. This corresponds to force-gradient algorithms of order 8. Combining the decomposition method with an advanced composition technique allows to increase the overall order up to a value of 16. The implementation and application of the introduced algorithms to numerical integration of the equations of motion in classical and quantum systems is considered as well. As is predicted theoretically and confirmed in molecular dynamics and celestial mechanics simulations, some of the new algorithms are particularly outstanding. They lead to much superior integration in comparison with known decomposition schemes such as the Verlet, Forest–Ruth, Chin, Suzuki, Yoshida, and Li integrators.