Related papers: Symplectic integrators in the shearing sheet
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,…
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.…
We construct a symplectic, globally defined, minimal-coordinate, equivariant integrator on products of 2-spheres. Examples of corresponding Hamiltonian systems, called spin systems, include the reduced free rigid body, the motion of point…
Calculating the long term solution of ordinary differential equations, such as those of the $N$-body problem, is central to understanding a wide range of dynamics in astrophysics, from galaxy formation to planetary chaos. Because generally…
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,…
Dependable numerical results from long-time simulations require stable numerical integration schemes. For Hamiltonian systems, this is achieved with symplectic integrators, which conserve the symplectic condition and exactly solve for the…
Hybrid symplectic integrators such as MERCURY are widely used to simulate complex dynamical phenomena in planetary dynamics that could otherwise not be investigated. A hybrid integrator achieves high accuracy during close encounters by…
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…
We implement and investigate the numerical properties of a new family of integrators connecting both variants of the symplectic Euler schemes, and including an alternative to the classical symplectic mid-point scheme, with some additional…
Symplectic integrators are the preferred method of solving conservative $N$-body problems in cosmological, stellar cluster, and planetary system simulations because of their superior error properties and ability to compute orbital…
By combining a standard symmetric, symplectic integrator with a new step size controller, we provide an integration scheme that is symmetric, reversible and conserves the values of the constants of motion. This new scheme is appropriate for…
We propose to use the properties of the Lie algebra of the angular momentum to build symplectic integrators dedicated to the Hamiltonian of the free rigid body. By introducing a dependence of the coefficients of integrators on the moments…
Recently, our group developed explicit symplectic methods for curved spacetimes that are not split into several explicitly integrable parts, but are via appropriate time transformations. Such time-transformed explicit symplectic integrators…
Two specialized algorithms for the numerical integration of the equations of motion of a Brownian walker obeying detailed balance are introduced. The algorithms become symplectic in the appropriate limits, and reproduce the equilibrium…
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…
We present a detailed comparison of several integration schemes applied to the dynamic system consisting of a charged particle on the Kerr background endowed with the axisymmetric electromagnetic test field. In particular, we compare the…
The existence of explicit symplectic integrators for general nonseparable Hamiltonian systems is an open and important problem in both numerical analysis and computing in science and engineering, as explicit integrators are usually more…
Hamilton's equations of motion form a fundamental framework in various branches of physics, including astronomy, quantum mechanics, particle physics, and climate science. Classical numerical solvers are typically employed to compute the…
Context. Integrating the motion of stars in a smoothed potential is necessary in many stellar and galactic studies. Previous works have often used numerical integrators that alternate between linear drifts and velocity kicks (such as the…
Using a Newtonian model of the Solar System with all 8 planets, we perform extensive tests on various symplectic integrators of high orders, searching for the best splitting scheme for long term studies in the Solar System. These…