Explicit Extended Phase-space Symplect-like Method With Midpoint Permutations

After symplectic algorithm was put forward it get long-term development in the field of celestial mechanics.For the symplectic algorithm don't show secular errors in angular momentum and energy integrals,it can preserve a symplectic structure.Generally,symplectic methods can be classified into two categories.One is explicit symplecic method,another is implicit one.The former is often used to calculate separable Hamilton system,but its application in the inseparable Hamiltonian systems is difficult,then the implicit symplectic algorithm will begin to play an important,but the implicit symplectic algorithm needs a lot of time iterative calculation,which will be inefficient.People began to consider bring the benefits of explicit algorithm into inseparable Hamiltonian system.In this paper I 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.And then I generalize this method to nonconservative systems.For instance,second post-Newtonian Hamiltonian of spinning compact binaries including dissipative effects from the gravitational radiation reaction,damped harmonic oscillator and the orbital motion of a dust particle experiencing Poynting-Robertson drag.In the numerical simulation,the explicit symplectic-like integrators with the midpoint permutations are superior to those with the sequence two permutations of momenta and coordinates,as well as 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.