Related papers: Adjustment of force-gradient operator in symplecti…
We take into account the dynamics of three types of models of rotating galaxies in polar coordinates in a rotating frame. Due to non-axisymmetric potential perturbations, the angular momentum varies with time, and the kinetic energy depends…
Symplectic integration algorithms are well-suited for long-term integrations of Hamiltonian systems because they preserve the geometric structure of the Hamiltonian flow. However, this desirable property is generally lost when adaptive…
We show that a recently discovered fourth order symplectic algorithm, which requires one evaluation of force gradient in addition to three evaluations of the force, when iterated to higher order, yielded algorithms that are far superior to…
Dynamics of a charged particle in the canonical coordinates is a Hamiltonian system, and the well-known symplectic algorithm has been regarded as the de facto method for numerical integration of Hamiltonian systems due to its long-term…
A consequent approach is proposed to construct symplectic force-gradient algorithms of arbitrarily high orders in the time step for precise integration of motion in classical and quantum mechanics simulations. Within this approach the basic…
Symplectic integration algorithms have become popular in recent years in long-term orbital integrations because these algorithms enforce certain conservation laws that are intrinsic to Hamiltonian systems. For problems with large variations…
Efficient fourth order symplectic integrators are proposed for numerical integration of separable Hamiltonian systems H(p,q)=T(p)+V(q). Symmetric splitting coefficients with five to nine stages are obtained by higher order decomposition of…
In recent decades, there have been many attempts to construct symplectic integrators with variable time steps, with rather disappointing results. In this paper we identify the causes for this lack of performance, and find that they fall…
Symplectic integration methods based on operator splitting are well established in many branches of science. For Hamiltonian systems which split in more than two parts, symplectic methods of higher order have been studied in detail only for…
Explicit symplectic integrators have been important tools for accurate and efficient approximations of mechanical systems with separable Hamiltonians. For the first time, the article proposes for arbitrary Hamiltonians similar integrators,…
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…
Symplectic integrators evolve dynamical systems according to modified Hamiltonians whose error terms are also well-defined Hamiltonians. The error of the algorithm is the sum of each error Hamiltonian's perturbation on the exact solution.…
It has previously been shown that varying the numerical timestep during a symplectic orbital integration leads to a random walk in energy and angular momentum, destroying the phase space-conserving property of symplectic integrators. Here…
In this paper an approach is outlined. With this approach some explicit algorithms can be applied to solve the initial value problem of $n-$dimensional damped oscillators. This approach is based upon following structure: for any…
Symplectic integrators separate a problem into parts that can be solved in isolation, alternately advancing these sub-problems to approximate the evolution of the complete system. Problems with a single, dominant mass can use mixed-variable…
It is well known that symplectic integrators lose their near energy preservation properties when variable step sizes are used. The most common approach to combine adaptive step sizes and symplectic integrators involves the Poincar\'e…
We propose a new framework of Hessian-free force-gradient integrators that do not require the analytical expression of the force-gradient term based on the Hessian of the potential. Due to that the new class of decomposition algorithms for…
We present a class of symplectic integrators adapted for the integration of perturbed Hamiltonian systems of the form $H=A+\epsilon B$. We give a constructive proof that for all integer $p$, there exists an integrator with positive steps…
The usual explicit finite-difference method of solving partial differential equations is limited in stability because it approximates the exact amplification factor by power-series. By adapting the same exponential-splitting method of…
The evolution of any factorized time-reversible symplectic integrators, when applied to the harmonic oscillator, can be exactly solved in a closed form. The resulting modified Hamiltonians demonstrate the convergence of the Lie series…