# Splitting methods for time integration of trajectories in combined electric and magnetic fields.

@article{Knapp2015SplittingMF, title={Splitting methods for time integration of trajectories in combined electric and magnetic fields.}, author={Christian Knapp and Alexander Kendl and Antti Koskela and Alexander Ostermann}, journal={Physical review. E, Statistical, nonlinear, and soft matter physics}, year={2015}, volume={92 6}, pages={ 063310 } }

The equations of motion of a single particle subject to an arbitrary electric and a static magnetic field form a Poisson system. We present a second-order time integration method which preserves well the Poisson structure and compare it to commonly used algorithms, such as the Boris scheme. All the methods are represented in a general framework of splitting methods. We use the so-called ϕ functions, which give efficient ways for both analyzing and implementing the algorithms. Numerical…

## 17 Citations

Energy behaviour of the Boris method for charged-particle dynamics

- PhysicsBIT Numerical Mathematics
- 2018

The Boris algorithm is a widely used numerical integrator for the motion of particles in a magnetic field. This article proves near-conservation of energy over very long times in the special cases…

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

- Mathematics, PhysicsSIAM J. Numer. Anal.
- 2021

Under the maximal ordering scaling case, a novel energy-preserving splitting scheme with computational cost per step independent from the strength of the magnetic field is proposed and in fact for a class of Lie-Trotter type splitting schemes, a uniform and optimal error bound is established.

Closed-Form Solutions for the Trajectories of Charged Particles in an Exponentially Varying Magnetostatic Field

- PhysicsIEEE Transactions on Plasma Science
- 2019

We present a new reference solution for charged particle motion in a strongly inhomogeneous magnetostatic field. The solution describes both bound and unbound particle motion, which can be split into…

Study of adaptive symplectic methods for simulating charged particle dynamics

- PhysicsJournal of Computational Dynamics
- 2019

In plasma simulations, numerical methods with high computational efficiency and long-term stability are needed. In this paper, symplectic methods with adaptive time steps are constructed for…

Energy-preserving splitting methods for charged-particle dynamics in a normal or strong magnetic field

- PhysicsAppl. Math. Lett.
- 2022

Continuous-stage symplectic adapted exponential methods for charged-particle dynamics with arbitrary electromagnetic fields

- Mathematics
- 2022

. This paper is devoted to the numerical symplectic approximation of the charged- particle dynamics (CPD) with arbitrary electromagnetic ﬁelds. By utilizing continuous-stage methods and exponential…

Semi-discretization and full-discretization with optimal accuracy for charged-particle dynamics in a strong nonuniform magnetic field

- Computer ScienceArXiv
- 2022

A new class of uniformly accurate methods with simple scheme is formulated for the three dimensional CPD in maximal ordering case based on the strategy given for the two dimensional case.

Fundamental derivation of two Boris solvers and the Ge-Marsden theorem.

- PhysicsPhysical review. E
- 2021

This work shows that, for a constant magnetic field, both magnetic algorithms can be further modified so that their trajectories are exactly on the gyrocircle at finite time steps.

The anatomy of Boris type solvers and large time-step plasma simulations

- PhysicsArXiv
- 2021

This work shows that this second-order Boris solver is much more accurate then previously thought and that its trajectory remains close to the exact orbit in a combined nonuniform electric and magnetic field at time-steps greater than the cyclotron period.

## References

SHOWING 1-10 OF 18 REFERENCES

Classical Molecular Dynamics Simulation with the Velocity Verlet Algorithm at Strong External Magnetic Fields

- Physics
- 1999

We present a new method for incorporating arbitrarily strong static homogeneous external magnetic fields into molecular dynamics computer simulations. Conventional techniques dealing with magnetic…

On the Numerical Integration of Ordinary Differential Equations by Symmetric Composition Methods

- MathematicsSIAM J. Sci. Comput.
- 1995

Free Lie algebra theory gives simple formulae for the number of determining equations for a method to have a particular order, and a new, more accurate way of applying the methods thus obtained to compositions of an arbitrary first-order integrator is described and tested.

Single particle motion in a Penning trap: description in the classical canonical formalism

- Physics
- 1992

This paper aims at the development of methods for the calculation of the characteristic frequencies of a Penning trap, taking into account deviations of the actual geometry from the ideal one,…

Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations

- Physics
- 2004

An almost symmetric Strang splitting scheme for the construction of high order composition methods☆

- MathematicsJ. Comput. Appl. Math.
- 2014

Exponential integrators

- Mathematics, Computer ScienceActa Numerica
- 2010

The main intention in this article is to present the mathematics behind these methods, and derive error bounds that are independent of stiffness or highest frequencies in the system.

Why is Boris algorithm so good

- Computer Science
- 2013

It is shown that the Boris algorithm conserves phase space volume, even though it is not symplectic, making it an effective algorithm for the multi-scale dynamics of plasmas.

Functions of matrices - theory and computation

- Mathematics
- 2008

A thorough and elegant treatment of the theory of matrix functions and numerical methods for computing them, including an overview of applications, new and unpublished research results, and improved…