Construction And Application Of Energy-conserving Algorithm For Hamiltonian Systems

Energy is an important conserved quantity in a conservative Hamiltonian dynamical system,which determines the system's motion on a certain level.In solving the conservative Hamiltonian system,whether the energy of the system can be exactly grasped is the key to accurately analyze the dynamic properties of the system.This has prompted us to construct and develop an energy-conserving algorithm based on the conservative Hamiltonian system.In this paper,based on the discretization of the Hamilton canonical equation,the gradient of each canonical variable is rewritten into the form of multiple Hamiltonian differences and quotients and averaged multiple times,we have successfully constructed an energy-conserving algorithm for 8-dimensional Hamiltonian systems with second-order accuracy and no truncation error,which fills the gap of this type of algorithm in high-dimensional Hamiltonian systems.We applied the newly constructed second-order 8-dimensional energy-conserving algorithm to the disordered discrete nonlinear Schrodinger equation,the Fermi-Pasta-Ulam-? model,the post-Newtonian spin compact binaries to analysis the performance of the newly algorithm from various aspects such as energy error,orbit error,and computational cost,the relevant numerical results were compared with the same-order Runge-Kutta method,implicit midpoint symplectic method,and extended phase-space explicit symplectic-like integrators.By comparing the performance of different algorithms,we find that the second-order 8-dimensional energy-conserving algorithm has an absolute advantage in keeping the system's energy,the energy error calculated by it is extremely low and shows a long-term stable trend.Meanwhile,we use the numerical solution obtained by the second-order 8-dimensional energy-conserving algorithm to analyze the orbit dynamics of the Fermi-Pasta-Ulam-? model,and find the critical energy value of the system from order to chaos.In a post-Newtonian conservative system of compact binaries with one body spinning,we analyzed the relationship between the celestial orbital shape,precession,gravitational wave radiation and the initial eccentricity of the orbit with the help of the energy-conserving schems.Since the energy-conserving algorithm is a stable numerical integrator that does not contain any truncation error and strictly maintains the system's energy,so it is a reliable tool for satisfying some specific purposes on the preservation of energies in numerical simulations with much longer times,and it is worth to further development.