Explicit Symplectic-Like Integrators With Midpoint Permutations for Spinning Compact Binaries

Abstract

We refine the recently developed fourth-order extended phase space explicit symplectic-like methods for inseparable Hamiltonians using Yoshida’s triple product combined with a midpoint permuted map. The midpoint between the original variables and their corresponding extended variables at every integration step is readjusted as the initial values of the original variables and their corresponding extended ones at the next step integration. The triple-product construction is apparently superior to the composition of two triple products in computational efficiency. Above all, the new midpoint permutations are more effective in restraining the equality of the original variables and their corresponding extended ones at each integration step than the existing sequent permutations of momenta and coordinates. As a result, our new construction shares the benefit of implicit symplectic integrators in the conservation of the second post-Newtonian Hamiltonian of spinning compact binaries. Especially for the chaotic case, it can work well, but the existing sequent permuted algorithm cannot. When dissipative effects from the gravitational radiation reaction are included, the new symplectic-like method has a secular drift in the energy error of the dissipative system for the orbits that are regular in the absence of radiation, as an implicit symplectic integrator does. In spite of this, it is superior to the same-order implicit symplectic integrator in accuracy and efficiency. The new method is particularly useful in discussing the long-term evolution of inseparable Hamiltonian problems.

Speed-Dependent Adaptive Partitioning QM/MM for Displacement Damage Simulations

“…this paper similar to Luo, et al 74 Although the strong coupling between the two parts of the EPS by these parameters is considered undesirable in the original EP-S paper, 71 these parameters allow the use of time steps much larger than 10 −5 ps, which is needed for practical simulations.…”

“…This is due to that the QM/MM partitions of the two parts of the EPS may become different during the internal steps of the algorithm. I use a 1 = a 2 = 1/2, b 1 = b 2 = 1/2, a = b = 1/2 in this paper similar to Luo, et al 74 Although the strong coupling between the two parts of the EPS by these parameters is considered undesirable in the original EP-S paper, 71 these parameters allow the use of time steps much larger than 10 −5 ps, which is needed for practical simulations.…”

Speed-Dependent Adaptive Partitioning QM/MM for Displacement Damage Simulations

“…(9) due to its assumption of velocity-independent acceleration. The extended phase space (EPS) algorithm [71][72][73][74] is an explicit time integration algorithm applicable to non-Lagrangian EOMs such as Eq. ( 9).…”

A Novel Energy-conserving Scheme for Eight-dimensional Hamiltonian Problems

“…Symmetric methods (Quinlan & Tremaine 1990) are similar to the symplectic integrators and do not lead to a secular, unphysical energy drift. Extended phase-space methods (Pihajoki 2015;Liu et al 2016;Luo et al 2017;Li & Wu 2017) are also explicitly symplectic-like or are symmetric schemes. As a point to note, the socalled conservation of energies in these integrators does not mean that these algorithms strictly conserve the energy integrals without any truncation errors from a theoretical viewpoint, but that does mean that these algorithms show no secular drift in the energy errors from a numerical viewpoint.…”