Construction of a Second-order Six-dimensional Hamiltonian-conserving Scheme

Research has analytically shown that the energy-conserving implicit nonsymplectic scheme of Bacchini, Ripperda, Chen, and Sironi provides a first-order accuracy to numerical solutions of a six-dimensional conservative Hamiltonian system. Because of this, a new second-order energy-conserving implicit scheme is proposed. Numerical simulations of a galactic model hosting a BL Lacertae object and magnetized rotating black hole background support these analytical results. The new method with appropriate time steps is used to explore the effects of varying the parameters on the presence of chaos in the two physical models. Chaos easily occurs in the galactic model as the mass of the nucleus, the internal perturbation parameter, and the anisotropy of the potential of the elliptical galaxy increase. The dynamics of charged particles around the magnetized Kerr spacetime is easily chaotic for larger energies of the particles, smaller initial angular momenta of the particles, and stronger magnetic fields. The chaotic properties are not necessarily weakened when the black-hole spin increases. The new method can be used for any six-dimensional Hamiltonian problems, including globally hyperbolic spacetimes with readily available (3 + 1) split coordinates.

Adjustment of Force–Gradient Operator in Symplectic Methods

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.

A Fourth Order Symplectic and Conjugate-Symplectic Extension of the Midpoint and Trapezoidal Methods

The paper presents fourth order Runge–Kutta methods derived from symmetric Hermite–Obreshkov schemes by suitably approximating the involved higher derivatives. In particular, starting from the multi-derivative extension of the midpoint method we have obtained a new symmetric implicit Runge–Kutta method of order four, for the numerical solution of first-order differential equations. The new method is symplectic and is suitable for the solution of both initial and boundary value Hamiltonian problems. Moreover, starting from the conjugate class of multi-derivative trapezoidal schemes, we have derived a new method that is conjugate to the new symplectic method.

The Oscillation Numbers and the Abramov Method of Spectral Counting for Linear Hamiltonian Systems

In this paper we consider linear Hamiltonian differential systems which depend in general nonlinearly on the spectral parameter and with Dirichlet boundary conditions. For the Hamiltonian problems we do not assume any controllability and strict normality assumptions which guarantee that the classical eigenvalues of the problems are isolated. We also omit the Legendre condition for their Hamiltonians. We show that the Abramov method of spectral counting can be modified for the more general case of finite eigenvalues of the Hamiltonian problems and then the constructive ideas of the Abramov method can be used for stable calculations of the oscillation numbers and finite eigenvalues of the Hamiltonian problems.