Related papers: Variable Timestep Integrators for Long-Term Orbita…
We reconsider the variational derivation of symplectic partitioned Runge-Kutta schemes. Such type of variational integrators are of great importance since they integrate mechanical systems with high order accuracy while preserving the…
We present a simple choice of integration variables that can be used to exploit the near-integrable character of problems in celestial mechanics. The approach is based on the well-known principle of variation of parameters: instead of…
Hamiltonian systems are differential equations which describe systems in classical mechanics, plasma physics, and sampling problems. They exhibit many structural properties, such as a lack of attractors and the presence of conservation…
Symplectic integration of autonomous Hamiltonian systems is a well-known field of study in geometric numerical integration, but for non-autonomous systems the situation is less clear, since symplectic structure requires an even number of…
We introduce a class of fourth order symplectic algorithms that are ideal for doing long time integration of gravitational few-body problems. These algorithms have only positive time steps, but require computing the force gradient in…
We propose a fast integrator to a class of dynamical systems with several temporal scales. The proposed method is developed as an extension of the variable step size Heterogeneous Multiscale Method (VSHMM), which is a two-scale integrator…
Symplectic integrators constructed from Hamiltonian and Lie formalisms are obtained as symplectic maps whose flow follows the exact solution of a "sourrounded" Hamiltonian K = H + h^k H_1. Those modified Hamiltonians depends virtually on…
Many force-gradient explicit symplectic integration algorithms have been designed for the Hamiltonian $H=T (\mathbf{p})+V(\mathbf{q})$ with kinetic energy $T(\mathbf{p})=\mathbf{p}^2/2$ in the existing references. When the force-gradient…
Numerical methods that preserve geometric invariants of the system, such as energy, momentum or the symplectic form, are called geometric integrators. Variational integrators are an important class of geometric integrators. The general idea…
Leapfrog integration has been the method of choice in N-body simulations owing to its low computational cost for a symplectic integrator with second order accuracy. We introduce a new leapfrog integrator that allows for variable timesteps…
Symplectic integrators are a foundation to the study of dynamical $N$-body phenomena, at scales ranging from from planetary to cosmological. These integrators preserve the Poincar\'e invariants of Hamiltonian dynamics. The $N$-body…
A parallel implementation of coupled spin-lattice dynamics in the LAMMPS molecular dynamics package is presented. The equations of motion for both spin only and coupled spin-lattice dynamics are first reviewed, including a detailed account…
Reliable studies of the long-term dynamics of planetary systems require numerical integrators that are accurate and fast. The challenge is often formidable because the chaotic nature of many systems requires relative numerical error bounds…
We describe an algorithm for long-term planetary orbit integrations, including the dominant post-Newtonian effects, that employs individual timesteps for each planet. The algorithm is symplectic and exhibits short-term errors that are…
This paper describes a fourth-order integration algorithm for the gravitational N-body problem based on discrete Lagrangian mechanics. When used with shared timesteps, the algorithm is momentum conserving and symplectic. We generalize the…
A fixed time-step variational integrator cannot preserve momentum, energy, and symplectic form simultaneously for nonintegrable systems. This barrier can be overcome by treating time as a discrete dynamic variable and deriving adaptive…
Long-term stability studies of nonlinear Hamiltonian systems require symplectic integration algorithms which are both fast and accurate. In this paper, we study a symplectic integration method wherein the symplectic map representing the…
In this work, we present a symplectic integration scheme to numerically compute space debris motion. Such an integrator is particularly suitable to obtain reliable trajectories of objects lying on high orbits, especially geostationary ones.…
A new approach is developed to integrate numerically the equations of motion for systems of interacting rigid polyatomic molecules. With the aid of a leapfrog framework, we directly involve principal angular velocities into the integration,…
We present and compare different numerical schemes for the integration of the variational equations of autonomous Hamiltonian systems whose kinetic energy is quadratic in the generalized momenta and whose potential is a function of the…