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Explicit Integrators for the Magnetized Equations of Motion in Particle in Cell Codes

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Explicit Integrators for the Magnetized Equations of Motion in Particle in Cell Codes Explicit integrators for the magnetized equations of motion in Particle in Cell codes The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Patacchini, L., and I.H. Hutchinson. “Explicit time-reversible orbit integration in Particle In Cell codes with static homogeneous magnetic field.” Journal of Computational Physics 228.7 (2009): 2604-2615. © 2009 Elsevier Inc. As Published http://dx.doi.org/10.1016/j.jcp.2008.12.021 Publisher Academic Press Version Original manuscript Citable link http://hdl.handle.net/1721.1/58724 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Explicit integrators for the magnetized equations of motion in Particle in Cell codes L. Patacchini and I.H. Hutchinson Plasma Science and Fusion Center, MIT Cambridge, Massachusetts, 02139, USA Abstract The development of a new orbit integrator (called Cyclotronic integrator) par- ticularly suitable for magnetized Particle in Cell codes where the Larmor angular frequency Ω is larger than any other characteristic frequencies is presented. This second order scheme, shown to be symplectic when the background magnetic field is static and uniform, is a bridge between the well-known Boris push in the limit of small time steps (Ω∆t 1) and the Spreiter and Walter Taylor expansion algorithm in the opposite limit (Ω∆t 1). The Boris and the Cyclotronic inte- grators' performances are investigated in terms of linear stability, local accuracy and conservation properties; in both uniform and two-dimensional magnetic field geometries. It is shown that in a uniform magnetic field and provided Ω∆t ∼< 1, the Cyclotronic integrator necessarily outperforms the two other alternatives. 1 Introduction The Boris integration scheme [1], designed to solve the single particle equations of motion in electric and magnetic fields x_ = v (1) mv_ = Q (E + v ^ B) ; is perhaps the most widely used orbit integrator in explicit Particle In Cell (PIC) simulations of plasmas. In Eqs (1) x and v are the particle position and velocity, m its mass and Q its charge. The idea of the Boris integrator is to offset x and v by half a time-step (∆t=2), and update them alternatively using the following Drift and Kick operators: D(∆t) := x0 − x = ∆tv (2) Q v0 + v K(∆t) := v0 − v = ∆t [E(x0) + ^ B(x0)]: (3) m 2 Although seemingly implicit (the right hand side of Eq. (3) contains both v and v0, the velocities at the beginning and end of the step), K can easily be inverted and the 1 scheme is in practice explicit as will be discussed in more detail in Section 2.1. The reasons for Boris scheme's popularity are twofold. It must first be recognized that the algorithm is extremely simple to implement, and offers second order accuracy while requiring only one force (or field) evaluation per step. Other integrators such as the usual or midpoint second order Runge-Kutta [2] require two field evaluations per step, thus considerably increasing the computational cost. The second reason is that for stationary electric and magnetic fields, the errors on conserved quantities such as the energy or the canonical angular momentum when the system is axisymmetric, are bounded for an infinite time (The error on those quantities is second order in ∆t as is the scheme). Those conservation properties, usually observed on long-time simulations of periodic or quasi-periodic orbits, are characteristic of time- reversible integrators [3]. The Boris scheme has a weak point however; it requires a fine resolution of the Larmor angular frequency Ω = QjBj=m, typically Ω∆t ∼< 0:1 [4, 5]. After a brief review of the Boris push, we present a novel integrator subject to a much weaker Larmor constraint. This scheme, called Cyclotronic integrator, splits the equations of motion (1) by taking advantage of the fact that in a uniform magnetic field and zero electric field the particle trajectory has a simple analytic form. Using this method, it is shown that provided the Larmor frequency is much higher than any other characteristic frequencies of the problem, the time step is only limited by stability considerations, typically leading to Ω∆t ∼< 1. If the background magnetic field is uniform, in addition to being time-reversible the Cyclotronic integrator is shown to be symplectic [6]; in other words it preserves the geometric structure of the Hamiltonian flow. Those different schemes are benchmarked on test problems involving uniform and 2D magnetic fields. Spreiter and Walter [7] previously attempted to relax the Larmor constraint, and developed a \Taylor expansion algorithm" for the static and uniform magnetic field regimes that we compare with the Cylotronic integrator. While both integrators have almost identical short-term performances, the Taylor expansion algorithm suffers from non time-reversibility and unconditional unstability. We therefore show that it should be preferred to the Cyclotronic integrator only when Ω∆t ∼> 1. 2 Derivation of Boris and Cyclotronic integrators 2.1 Boris integrator The Boris integrator is a time-splitting method for Eqs (1): the equations of motion are separated in two parts that are successively integrated in a Verlet form, in other words x x (t + ∆t) = D(∆t=2) · K(∆t) · D(∆t=2) (t); (4) v v where the Drift and Kick operators (D and K) are defined in Eqs (2,3). If R∆' denotes a rotation of characteristic vector 1 ∆t B B ∆' = 2 tan− ( Ω) ∼ Ω∆t ; (5) 2 B B 2 K(∆t) := v ! v0 can be split in the following way [1] QE∆t v∗ = v + 2m K(∆t) := 8 v∗∗ = R∆'v∗ (6) > QE∆t < v0 = v∗∗ + 2m : Eqs (2,6) readily show that the Boris:> integrator is time-reversible. Indeed the Drift operator does not act on the particle velocity, and the Kick operator does not act on the position. In PIC codes it is customary to define the position and velocity with half a time-step of offset, which amounts to concatenating the two adjacent D(∆t=2) from successive steps in Eq. (4). A popular variant of this integrator (known as the \tan" transformation [1]), second B order in ∆t , consists in letting ∆' = Ω∆t B in Eq. (6). Regardless of the form used for ∆' however, the Drift operator (2) requires Ω∆t 1, which is a severe limitation if the other characteristic frequencies (Such as the quadrupole harmonic frequency !0 introduced in Section 3) are much smaller than Ω. 2.2 Symplectic and time-reversible integration Time-reversible integrators contain the subclass of symplectic schemes, which has re- cieved considerable attention in the last decades in particular in connection with as- trodynamics [8] and accelerator physics [9]. The fundamental idea behind symplectic integration of (systems of) Ordinary Dif- ferential Equations (ODEs) is to ensure that the chosen scheme is a canonical map, in other words that there exists canonical coordinates (q; p) related to the physical variables (x; v) such that the flow Z(τ) = (q; p)(τ) derives from a Hamiltonian H:~ dp dq = −∇qH~ = rpH~ ; (7) dt dt in which case there exists a Liouville operator ΨH~ such that dZ = fZ; H~ (Z)g = Ψ Z (8) dt H~ or equivalently 8τ 2 R z(τ) = eτΨH~ Z(0); (9) where f:; :g stands for the Poisson bracket. Indeed if the original ODEs derive from a Hamiltonian H(q; p), one can show that the Hamiltonian from which the flow of a consistent nth order symplectic integrator derives takes the form: H(~ q; p) = H(q; p) + δH(q; p; ∆t), where δH = O(∆tn) [10]. Because the integrator exactly preserves H~ and its integral invariants, it is expected to conserve slightly modified expressions of the integral invariants of H. Hence no secular drift in the original problem's energy or integral invariants is to occur. For a more complete introduction on symplectic integration avoiding unnecessary mathematical formalism, the reader is referred to Ref. [6]. 3 The Boris integrator is known for its outstanding conservation properties. However as pointed out by Stolz et al. [11], there is no guarantee that it is symplectic. It is nonetheless time-reversible, and it has been shown under very reasonable assumptions that this condition is sufficient to explain the absence of secular drift in the conserved quantities, provided the orbit we integrate is periodic or quasi-periodic [3]. 2.3 Cyclotronic integrator The time independent Hamiltonian for single particle motion in the presence of a uniform background magnetic field B = Bez can easily be written in cylindrical coor- dinates: p2 p2 1 p 2 H(q; p) = ρ + z + ' − QA (ρ, z) + Qφ(q); (10) 2m 2m 2m ρ ' where the generalized momentum p is given by pz = mz_ p = mρ_ 8 ρ (11) 2 A' <> p' = mρ '_ + Q mρ and q = (z; ρ, '). The vector:>potential A satisfies r ^ A = Bez and is chosen to be Bρ A = 2 e', while E = −∇φ. The flow deriving from the full Hamiltonian H in Eq. (10) is not integrable. It is however possible to rewrite H as H = H1 + H2 where the flows associated with H1;2 are exactly integrable for any time-step ∆t as follows: 2 2 2 pρ pz 1 p' • Drift part: H1(q; p) = 2m + 2m + 2m ρ − QA'(ρ, z) B Uniform helical motion around B with angle ∆' = Ω∆t B . • Kick part: H2(q; p) = Qφ(q) Momentum increase of vector −Qrφ∆t. Using the Baker Campbell Hausdorff formula [6], one can show that e∆tΨH = e(∆t=2)ΨH1 · e∆tΨH2 · e(∆t=2)ΨH1 + O(∆t3) (12) A second order symplectic integrator for H is therefore A(∆t) = e(∆t=2)ΨH1 · e∆tΨH2 · e(∆t=2)ΨH1 (13) or its conjugate A¯(∆t) = e(∆t=2)ΨH2 · e∆tΨH1 · e(∆t=2)ΨH2 .
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