Efficient energy-preserving methods for charged-particle dynamics

In this paper, energy-preserving methods are formulated and studied for solving charged-particle dynamics. We first formulate the scheme of energy-preserving methods and analyze its basic properties including algebraic order and symmetry. Then it is shown that these novel methods can exactly preserve the energy of charged-particle dynamics. Moreover, the long time momentum conservation is studied along such energy-preserving methods. A numerical experiment is carried out to illustrate the notable superiority of the new methods in comparison with the popular Boris method in the literature

Error estimates of some splitting schemes for charged-particle dynamics under strong magnetic field

In this work, we consider the error estimates of some splitting schemes for the charged-particle dynamics under a strong magnetic field. We first propose a novel energy-preserving splitting scheme with computational cost per step independent from the strength of the magnetic field. Then under the maximal ordering scaling case, we establish for the scheme and in fact for a class of Lie-Trotter type splitting schemes, a uniform (in the strength of the magnetic field) and optimal error bound in the position and in the velocity parallel to the magnetic field. For the general strong magnetic field case, the modulated Fourier expansions of the exact and the numerical solutions are constructed to obtain a convergence result. Numerical experiments are presented to illustrate the error and energy behaviour of the splitting schemes.Comment: 26 pages, 4 figure

Discrete line integral method for the Lorentz force system

In this paper, we apply the Boole discrete line integral to solve the Lorentz force system which is written as a non-canonical Hamiltonian system. The method is exactly energy-conserving for polynomial Hamiltonians of degree $\nu \leq 4$. In any other case, the energy can also be conserved approximatively. With comparison to well-used Boris method, numerical experiments are presented to demonstrate the energy-preserving property of the method

Explicit non-canonical symplectic algorithms for charged particle dynamics

We study the non-canonical symplectic structure, or K-symplectic structure inherited by the charged particle dynamics. Based on the splitting technique, we construct non-canonical symplectic methods which is explicit and stable for the long-term simulation. The key point of splitting is to decompose the Hamiltonian as four parts, so that the resulting four subsystems have the same structure and can be solved exactly. This guarantees the K-symplectic preservation of the numerical methods constructed by composing the exact solutions of the subsystems. The error convergency of numerical solutions is analyzed by means of the Darboux transformation. The numerical experiment display the long-term stability and efficiency for these methods.Comment: 9 pages,6 figure

Explicit high-order symplectic integrators for charged particles in general electromagnetic fields

This article considers non-relativistic charged particle dynamics in both static and non-static electromagnetic fields, which are governed by nonseparable, possibly time-dependent Hamiltonians. For the first time, explicit symplectic integrators of arbitrary high-orders are constructed for accurate and efficient simulations of such mechanical systems. Performances superior to the standard non-symplectic method of Runge-Kutta are demonstrated on two examples: the first is on the confined motion of a particle in a static toroidal magnetic field used in tokamak; the second is on how time-periodic perturbations to a magnetic field inject energy into a particle via parametric resonance at a specific frequency.Comment: Submitted to JCP on May 13, 201

Arbitrarily high-order energy-preserving methods for simulating the gyrocenter dynamics of charged particles

Gyrocenter dynamics of charged particles plays a fundamental role in plasma physics. In particular, accuracy and conservation of energy are important features for correctly performing long-time simulations. For this purpose, we here propose arbitrarily high-order energy conserving methods for its simulation. The analysis and the efficient implementation of the methods are fully described, and some numerical tests are reported.Comment: 23 pages, 4 figure

Bill2d - a software package for classical two-dimensional Hamiltonian systems

We present Bill2d, a modern and efficient C++ package for classical simulations of two-dimensional Hamiltonian systems. Bill2d can be used for various billiard and diffusion problems with one or more charged particles with interactions, different external potentials, an external magnetic field, periodic and open boundaries, etc. The software package can also calculate many key quantities in complex systems such as Poincar\'e sections, survival probabilities, and diffusion coefficients. While aiming at a large class of applicable systems, the code also strives for ease-of-use, efficiency, and modularity for the implementation of additional features. The package comes along with a user guide, a developer's manual, and a documentation of the application program interface (API)

Magnetic Hamiltonian Monte Carlo

Hamiltonian Monte Carlo (HMC) exploits Hamiltonian dynamics to construct efficient proposals for Markov chain Monte Carlo (MCMC). In this paper, we present a generalization of HMC which exploits \textit{non-canonical} Hamiltonian dynamics. We refer to this algorithm as magnetic HMC, since in 3 dimensions a subset of the dynamics map onto the mechanics of a charged particle coupled to a magnetic field. We establish a theoretical basis for the use of non-canonical Hamiltonian dynamics in MCMC, and construct a symplectic, leapfrog-like integrator allowing for the implementation of magnetic HMC. Finally, we exhibit several examples where these non-canonical dynamics can lead to improved mixing of magnetic HMC relative to ordinary HMC.Comment: 34th International Conference on Machine Learning (ICML 2017

Explicit symplectic adapted exponential integrators for charged-particle dynamics in a strong and constant magnetic field

This paper studies explicit symplectic adapted exponential integrators for solving charged-particle dynamics in a strong and constant magnetic field. We first formulate the scheme of adapted exponential integrators and then derive its symplecticity conditions. Based on the symplecticity conditions, we propose five practical explicit symplectic adapted exponential integrators. Two numerical experiments are carried out and the numerical results demonstrate the remarkable numerical behavior of the new methods.Comment: The content of this manuscript is included in a new paper of ours as one section. We will submit the new paper to arXi

Performance of the BGSDC integrator for computing fast ion trajectories in nuclear fusion reactors

Modelling neutral beam injection (NBI) in fusion reactors requires computing the trajectories of large ensembles of particles. Slowing down times of up to one second combined with nanosecond time steps make these simulations computationally very costly. This paper explores the performance of BGSDC, a new numerical time stepping method, for tracking ions generated by NBI in the DIII-D and JET reactors. BGSDC is a high-order generalisation of the Boris method, combining it with spectral deferred corrections and the Generalized Minimal Residual method GMRES. Without collision modelling, where numerical drift can be quantified accurately, we find that BGSDC can deliver higher quality particle distributions than the standard Boris integrator at comparable cost or comparable distributions at lower cost. With collision models, quantifying accuracy is difficult but we show that BGSDC produces stable distributions at larger time steps than Boris.Comment: New version has multiple updates, clarifications in the text and new figure