Extension Of Force-gradient Symplectic Algorithm And Its Application To Chaotic Spiral Arms Of Galaxies

Symplectic algorithm is an ideal tool for studying the long-term evolution of Hamiltonian systems because of its characteristic of preserving symplectic and energy.However,symplectic algorithms higher than the second order all contain negative time coefficients and are not suitable for time irreversible systems.In order to remedy this defect,the time coefficients in the algorithm are all positive,many force-gradient explicit symplectic integration algorithms have been designed for the Hamiltonian H=T(p)+V(q)with kinetic energy T(p)=p2/2 in the existing references.When the force-gradient operator in the force-gradient explicit symplectic integration algorithm is properly adjusted to make it a new operator,force-gradient explicit symplectic integration algorithms using the new operator will apply for a class of Hamiltonian problems H=K(p,q)+V(q)with integrable part(?),where aij=aij(q)and bi=bi(q)are functions of coordinates q.The newly adjusted operator is no longer a force-gradient operator,but a momentum-type operator similar to that associated with potential energy V.The newly extended(or adjusted)algorithms are no longer solvers of the original Hamiltonian,but solvers of slightly modified Hamiltonian,which are also explicit symplectic algorithms with symmetry or time reversibility.Numerical tests show that the standard symplectic integrators without the new operator are generally poorer than the corresponding extended methods with the new operator in computational accuracies and efficiencies.The optimized methods have better accuracies than the corresponding non-optimized counterparts.Among the tested symplectic methods,the two extended optimized seven-stage fourth-order methods of Omelyan,Mryglod and Folk exhibit the best numerical performance.As a result,one of the two optimized algorithms is used to study the orbital dynamical features of a modified Hénon-Heiles system and a spring pendulum.These extended integrators allow for integrations in Hamiltonian problems,such as the spiral structure in self-consistent models of rotating galaxies and the spiral arms in galaxies.In this paper,chaotic spiral arms in the galaxy model are studied by using the newly extended force-gradient symplectic algorithm.In order to study the chaotic dynamics characteristics of the same chaotic spiral arms of galaxies(A19,B19 and C19)under different conditions of n(n is the number of so-called radial terms in potential series expansion),we change the number of n,that is to say,when n=19,We have the original models "A19","B19" and"C19";when n=9,there are models "A9","B9" and "C9";When n=4,we have "A4","B4" and "C4".And we know the step size τ=5 × 10 4Thmct(The half-mass crossing time Thmct=Thub/300,Hubble time Thub=1/82.4)is a good choice.However,in this step size case,the calculation time is very long,and the accuracy of each algorithm is basically the same.Therefore,in order to improve the computational efficiency of the chaotic spiral arms of galaxies,we use a large time step τ=0.5*Thmct.And in order to find a more suitable algorithm,we use several algorithms to compare the accuracy.Among the methods tested,N4P has the best numerical accuracy.Therefore,the N4P method is used to study the orbital dynamics of A19,A9,A4,B19,B9,B4,C19,C9 and C4 respectively.The results show that in the "A" series models,"B" series models and "C" series models,the smaller the value of n,the weaker the chaos of the chaotic spiral arms.The larger r value is,the weaker the overall chaos intensity of the system is.