Abstract
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 a force–gradient operator is appropriately adjusted as a new operator, it is still suitable for a class of Hamiltonian problems H=K(p,q)+V(q) with integrable part K(p,q)=∑i=1n∑j=1naijpipj+∑i=1nbipi, where aij=aij(q) and bi=bi(q) are functions of coordinates q. The newly adjusted operator is not a force–gradient operator but is similar to the momentum-version operator associated to the potential V. The newly extended (or adjusted) algorithms are no longer solvers of the original Hamiltonian, but are solvers of slightly modified Hamiltonians. They are explicit symplectic integrators 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.
Highlights
According to the perturbation decomposition of Hamiltonian systems, a number of higher order explicit symplectic integrations were developed and generalized in some references [25,26]. It can be seen from the above presentations that the construction of symplectic integrators is closely related to Hamiltonian systems or their splitting forms
The above demonstrations show that the third-order truncation error term C in the second-order method M2 and B belong to momentum-version operators
Many force–gradient explicit symplectic integration algorithms with the force–gradient operator C for the Hamiltonian (2) with the kinetic energy (1) have been in [21,22,23]. These algorithms become useless for the Hamiltonian (18) with the integrable kinetic energy (19) if the force–gradient operator C is not altered
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Ruth [3] proposed second- and third-order explicit symplectic methods for Hamiltonian systems of the form H = T (p) + V (q) Along this direction, higher order standard explicit symplectic schemes were developed by many authors [17,18,19,20]. When dealing with Hamiltonian systems with the integrable perturbation decomposition form H = H0 (p, q) + eH1 (q), the Wisdom-Holman symplectic map of second order [24] drastically improves the numerical accuracy, compared with the standard explicit symplectic method of second order for the Hamiltonian splitting form. According to the perturbation decomposition of Hamiltonian systems, a number of higher order explicit symplectic integrations were developed and generalized in some references [25,26] It can be seen from the above presentations that the construction of symplectic integrators is closely related to Hamiltonian systems or their splitting forms.
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