Multiple Boris integrators for particle-in-cell simulation

Seiji Zenitania,b,∗, Tsunehiko N. Katoc

aResearch Center for Urban Safety and Security, Kobe University, 1-1 Rokkodai-cho,Nada-ku, Kobe 657-8501, Japan

bResearch Institute for Sustainable Humanosphere, Kyoto University, Gokasho, Uji,Kyoto 611-0011, Japan

cCenter for Computational Astrophysics, National Astronomical Observatory of Japan,2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan

Abstract

We construct Boris-type schemes for integrating the motion of charged par-

ticles in particle-in-cell (PIC) simulation. The new solvers virtually combine

the 2-step Boris procedure arbitrary n times in the Lorentz-force part, and

therefore we call them the multiple Boris solvers. Using Chebyshev polyno-

mials, a one-step form of the new solvers is provided. The new solvers give

n2 times smaller errors, allow larger timesteps, and have a long-term stabil-

ity. We present numerical tests of the new solvers, in comparison with other

particle integrators.

Keywords: Boris integrator; Particle-in-cell simulation; Lorentz force

equation

1. Introduction

Particle-in-cell (PIC) simulation [1, 2] has been extensively used to study

complex kinetic phenomena in plasma systems. The Boris solver [3] (or the

∗Corresponding author.E-mail address: zenitani@port.kobe-u.ac.jp

Preprint submitted to Computer Physics Communications October 8, 2019

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Boris pusher) is a de fact standard method to advance charged particles in

PIC simulation. Although its algorithm is simple, it is robust and provides

moderately good accuracy. For these reasons, it has been used about a half

century.

Since mid 2010s, there is a renewed interest to develop new particle in-

tegrators. In recent PIC simulations, one often solves the system evolution

much longer than before. In such a case, we want to solve the particle mo-

tion as accurately as possible, because small errors could be accumulated

over many timesteps. The Boris solver is proven to be stable [4, 5], however,

it does produce a second-order error. For this reason, there is a growing

demand for higher-accuracy extensions to the Boris solver.

Umeda [6, 7] has proposed multistep Boris solvers. Repeating a Boris-type

procedure multiple steps in the Lorentz-force part, the author managed to set

an effective timestep 2N−2 times smaller. Since the Boris solver is a second-

order scheme, N -step Boris solvers provide 4N−2 times higher accuracy for

gyration. Zenitani & Umeda [8] have proposed to use a rotation formula in

the Lorentz-force part. The solver virtually provides no error in the phase

angle in gyration. Because of operator-splitting, their scheme offers second-

order accuracy.

In addition to higher numerical accuracy, it is also important to improve

computation cost. A fast solver helps us to reduce the total computation

time of PIC simulation. The cost depends on many other factors, but the

timestep ∆t is certainly one of them. If the solver allows us to use a larger

∆t, we can further reduce the computation time. Thus it is important to

evaluate the constraint on ∆t for particle solvers.

2

In this article, we propose new Boris-type integrators to solve the charged-

particle motion. The new solvers use a Boris-type procedure multiple times in

the Lorentz-force part. We call them “multiple Boris solvers.” They provide

higher accuracy and allow larger timesteps than the original Boris solver. In

Section 2, we briefly review the classical 2-step Boris solver and discuss its

graphical meanings. In Section 3, we introduce our multiple Boris solvers.

Using Chebyshev polynomials, we derive coefficients for n-tuple Boris solvers,

where n is an arbitrary integer. Then we discuss errors, limitations to the

timesteps, and the stability of the phase-space volume. In Section 4, we

present benchmark results. The new solvers and other particle integrators

are briefly compared. Section 5 contains discussion and summary.

2. Boris solver

We describe a popular form of the Boris algorithm [1, 2]. We refer to it

as the classical Boris solver. The algorithm solves the particle motion in a

time-staggered manner,

xt+∆t − xt

∆t=

ut+ ∆t2

γt+∆t2

, (1)

mut+ ∆t

2 − ut−∆t2

∆t= q

(Et +

ut+ ∆t2 + ut−∆t

2

2γt×Bt

). (2)

The superscripts (t, t + ∆t, ...) indicate time, u = γv is the spatial part of

a 4-vector, and γ = [1 − (v/c)2]−1/2 = [1 + (u/c)2]1/2 is the Lorentz factor.

Other symbols have their standard meanings. Hereafter we denote Bt as B

for brevity.

The acceleration part (Eq. (2)) is split into the Coulomb-force part for

3

the first half:

u− = ut−∆t2 + ε∆t, (3)

the middle Lorentz-force part:

t ≡ θ

2b̂ =

ωc∆t

2b̂ =

q∆t

2mγ−B, (4a)

u′ = u− + u− × t, (4b)

u+ = u− +2

1 + t2

(u′ × t

), (4c)

and the Coulomb-force part for the second half:

ut+ ∆t2 = u+ + ε∆t. (5)

In these equations, ε = (q/2m)Et, ωc = (qB/mγ−) is the gyro frequency,

θ = 2t = ωc∆t is the gyration angle during ∆t, b̂ = B/|B| is a unit vector

in the B direction, u− and u+ are two intermediate states.

We focus on the Lorentz-force part in the middle (Eqs. (4a)–(4c)). This

2-step procedure is illustrated in Figure 1(a). The coefficient 2/(1 + t2) is

provided such that the particle energy remains constant in the Lorentz-force

part, γ− = γ+ = γt. A base angle α is given by,

α ≡ arctan t (6)

We will use the following relations for α in subsequent discussions.

sinα =t√

1 + t2, cosα =

1√1 + t2

, tanα = t (7)

sin 2α =2t

1 + t2, cos 2α =

1− t21 + t2

(8)

4

From Eqs. (4b), (4c), and (8), we obtain

u+ =1

1 + t2

((1− t2)u− + 2(u− × t) + 2(u− · t)t

)(9)

= u− cos 2α + (u− × b̂) sin 2α + (1− cos 2α)u−‖ (10)

= (u− − u−‖ ) cos 2α + (u− × b̂) sin 2α + u−‖ , (11)

where the subscript ‖ indicates the component along B. These forms are

known as Euler–Rodrigues rotation formula. The equations correspond to

a rotation about the magnetic field by an approximate angle of 2α, instead

of the true angle θ. As indicated in Figure 1(a), the half acceleration in

the tangent direction (u−× t; Eq. (4b)) corresponds to the half approximate

angle α by the Boris solver. The true rotation angle θ contains a second-order

error of δθ/θ ≈ − 112θ2.

θ ≈ 2α = 2 arctanθ

2= θ

(1− 1

3

(θ2

)2

+1

5

(θ2

)4

− · · ·). (12)

3. Multiple Boris solver

Umeda [6] has proposed a novel way to reduce the error (δθ/θ). He

employed a half timestep ∆t → 12∆t in the Lorentz-force part (Eqs. (4a)–

(4c)) and then repeated the Boris procedure twice. Importantly, the timestep

for the position (Eq. (1)) and the Coulomb-force parts (Eqs. (3) and (5)) are

unchanged. We call this method the double Boris solver, and it is illustrated

in Figure 1(b). Instead of Eq. (4a), we use

t ≡ θ

4b̂ =

q∆t

4mγ−B, α ≡ arctan t (13)

5

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(a) Boris solver (b) Double Boris solver

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u0<latexit sha1_base64="2D/uO79zanxHoZtPQ1qM2buUv7E=">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</latexit><latexit sha1_base64="2D/uO79zanxHoZtPQ1qM2buUv7E=">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</latexit><latexit sha1_base64="2D/uO79zanxHoZtPQ1qM2buUv7E=">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</latexit><latexit sha1_base64="2D/uO79zanxHoZtPQ1qM2buUv7E=">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</latexit>

u0<latexit sha1_base64="2D/uO79zanxHoZtPQ1qM2buUv7E=">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</latexit><latexit sha1_base64="2D/uO79zanxHoZtPQ1qM2buUv7E=">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</latexit><latexit sha1_base64="2D/uO79zanxHoZtPQ1qM2buUv7E=">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</latexit><latexit sha1_base64="2D/uO79zanxHoZtPQ1qM2buUv7E=">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</latexit>

u+exact

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u+exact

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u00<latexit sha1_base64="GlPJeIMadP0khNZ5/LSf+Ehz1aE=">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</latexit><latexit sha1_base64="GlPJeIMadP0khNZ5/LSf+Ehz1aE=">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</latexit><latexit sha1_base64="GlPJeIMadP0khNZ5/LSf+Ehz1aE=">AAADX3ichVI9T9tQFD2JKVBoQ4ClqEtEFKVT9AJIwIbKwkgFgSASRbbzGqz4S7ZjhCz/ga5dyw9A/JH+gQ4dO3ZmYGHo8YszRCjNs2zfj3Puve/ea/i2FUZC/C4UtYU3i0vLb1dW370vrZXXNy5CbxSYsmV6the0DT2UtuXKVmRFtmz7gdQdw5aXxvA481/GMggtzz2P7nzZdfSBa321TD2i6aoTSzMZpfV6r1wVjb3DA7F7WHktNBtCnSryc+qtFyrooA8PJkZwIOEiomxDR8jnGk0I+LR1kdAWULKUXyLFCrkjoiQROq1DfgfUrnOrSz2LGSq2ySw234DMCmril3gQT+KneBR/xcvMWImKkdVyx78x5kq/t/btw9kzWbN5BvEpasQbKnOfkkNvhJu52Sa4//EDdUOJW3UzR/lcohL6YtpNsrP+dZVlEmNyj0xLUKU/zTs5K9aAmXTKqdKynhpqPuncKvqcVfyKl03cVDOP5mQOcsw0v6N2YDxX8rlzk8WqzBYudhpNyl/2qkef8+1bxkds4xM7sI8jnOAULdbg4Dt+4L74R1vSSlp5DC0Wcs4mpo629Q90jbMN</latexit><latexit sha1_base64="GlPJeIMadP0khNZ5/LSf+Ehz1aE=">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</latexit>

Figure 1: (a) classical Boris solver and (b) double Boris solver.

The base angle α is redefined, as indicated in Figure 1(b). Then, we repeat

the 2-step Boris procedure (Eqs. (4b) and (4c)) twice. We obtain a half state

u′′ by the first Boris procedure, which is given by Eq. (11) with Eq. (13).

Applying ·b̂ to Eq. (11), one can confirm that the parallel component is

preserved, u′′‖ = (u′′ · b̂)b̂ = u−‖ . Substituting u′′ to Eq. (11) again, we obtain

the next state u+,

u+ = (u′′ − u−‖ ) cos 2α + (u′′ × b̂) sin 2α + u′′‖ (14)

= (u− − u−‖ ) cos 4α + (u− × b̂) sin 4α + u−‖ (15)

The second Boris operation advances the phase angle by another 2α, and

the entire procedure gives gyration by 4α. In Ref. [6], the author proposed

a 3-step procedure to repeat the Boris operation twice. Now we obtain a

6

smaller second-order error in gyration, δθ/θ ≈ − 148θ2.

θ ≈ 4α = 4 arctanθ

4= θ

(1− 1

3

(θ4

)2

+1

5

(θ4

)4

− · · ·). (16)

Let us extend this idea. In order to better approximate the gyro motion,

we propose to repeat the Boris procedure n times, where n is an arbitrary

positive integer. We call the resulting schemes “multiple Boris solvers.” Here

below, we describe its procedure in the Lorentz-force part.

We define the tangent vector t and the base angle α as follows.

t ≡ θ

2nb̂ =

q∆t

2nmγ−B, α ≡ arctan t. (17)

Note that t is divided by n. From the discussion on the double Boris solver,

repeating the Boris operation n times, one can obtain a rotation angle of

2nα. The next state u+n should be given by

u+n = (u− − u−‖ ) cos(2nα) + (u− × b̂) sin(2nα) + u−‖ (18)

The subscript n indicates quantities for the n-tuple solver. For brevity, we

hereafter omit the minus sign from u−. Since we desire a simulation-friendly

expression with t, we modify Eq. (18) to

u+n = cos(2nα)u +

sin(2nα)

t(u× t) +

2 sin2(nα)

t2(u · t)t (19)

= cn1u + cn2 (u× t) + cn3(u · t)t (20)

where cn1, cn2, and cn3 are coefficients.

Next, we construct generic expressions for these coefficients. We use

Chebyshev polynomials of the first kind Tn(x) : T0(x) = 1, T1(x) = x, T2(x) =

2x2 − 1, . . . and the second kind Un(x) : U0(x) = 1, U1(x) = 2x, U2(x) =

7

4x2 − 1, . . . . They satisfy useful relations:

cosnx = Tn(cosx), sinnx = sinx Un−1(cosx) (21)

We also employ

p ≡ cos 2α =1− t21 + t2

, 1 + p =sin 2α

t=

2

1 + t2. (22)

With help from Eqs. (7), (8), (21), and (22), we express the coefficients in

Eq. (20) in the following way,

cn1 = Tn(cos 2α) = Tn(p) (23)

cn2 =1

tsin 2α Un−1(cos 2α) = (1 + p)Un−1(p) (24)

cn3 =

1 + p (for n = 1)

(1 + p)(Uk(p) + Uk−1(p)

)2

(for n = 2k + 1)

2

((1 + p)Uk−1(p)

)2

(for n = 2k)

(25)

Here, k is a positive integer. We derive Eq. (25) for n = 2k+1 (n = 3, 5, . . . )

in Appendix A. Practically, we compute p first, and then calculate Eqs. (23)-

(25) by using p. Some higher-order polynomials can be calculated via nesting

relations [9]:

Tmn(x) = Tm(Tn(x)), Umn−1(x) = Um−1(Tn(x))Un−1(x) (26)

8

We can also express the coefficients with t. From the polynomials,

cn1 = Tn

(1− t21 + t2

)(27)

cn2 =2

1 + t2Un−1

(1− t21 + t2

)(28)

cn3 =

2

1+t2(for n = 1)

21+t2

(Uk

(1−t21+t2

)+ Uk−1

(1−t21+t2

))2

(for n = 2k + 1)

8(1+t2)2

(Uk−1

(1−t21+t2

))2

(for n = 2k)

(29)

we obtain explicit forms, for example,

c11 =1− t21 + t2

, c12 =2

1 + t2, c13 =

2

1 + t2, (30)

c21 =1− 6t2 + t4

(1 + t2)2, c22 =

4(1− t2)

(1 + t2)2, c23 =

8

(1 + t2)2, (31)

c31 =1− 15t2 + 15t4 − t6

(1 + t2)3,

c32 =2(t2 − 3)(3t2 − 1)

(1 + t2)3, c33 =

2(t2 − 3)2

(1 + t2)3, (32)

c41 =

((1− t2)2 − 24t2

)(1− t2)2 + 16t4

(1 + t2)4

=1− 28t2 + 70t4 − 28t6 + t8

(1 + t2)4,

c42 =8(1− t2)(1− 6t2 + t4)

(1 + t2)4, c43 =

32(1− t2)2

(1 + t2)4(33)

It is interesting to see that the cn1, cn2, and cn3 share the same denominator,

(1 + t2)n. One can confirm this by inspecting Eqs. (27)–(29).

As a result, the multiple Boris solver advances the velocity by using

Eqs. (3), (17), (22)–(25), (20), and (5). One can also use the coefficients

in Eqs. (30)–(33) instead of Eqs. (22)–(25).

9

(a)

(b)

Figure 2: (a) Relative accuracy in phase angle and (b) relative length of Larmor radius

for the multiple Boris solvers.

10

The Lorentz-force part of the multiple Boris solvers provides a second-

order error in phase angle. From Eq. (17), we obtain

θ ≈ 2nα = 2n arctanθ

2n= θ

(1− θ2

12n2+

θ4

80n4− · · ·

). (34)

Thus, the multiple Boris solvers provide n2 times smaller errors than the

single Boris solver. Eq. (34) approaches the exact angle for n→∞.

In Figure 2(a), we plot relative accuracy in phase angle, θ−1·2n arctan(θ/2n),

for several n. In the classical Boris solver, which is equivalent to the n = 1

case, we typically set θ = ωc∆t < 0.2–0.3 to keep an error within 1%. Con-

sidering θ2/(12n2) . 0.01, we obtain

θ . 0.35n. (35)

The multiple Boris solvers allow n-times larger timestep than the single Boris

solver.

Next, we evaluate an effective Larmor radius. We consider a simple gy-

ration about B in the case of E = 0. From Eq. (1), we obtain

∣∣xt+∆t − xt∣∣ =

∣∣ut+ ∆t2

∣∣γt+

∆t2

∆t = |vR|∆t, (36)

where vR is the velocity. The two points xt+∆t and xt are located on the same

circle with a Larmor radius of r′L. The two points should be two endpoints

of an arc with a central angle 2nα. Then we expect

|xt+∆t − xt| = 2r′L sin(2nα

2

). (37)

From Eqs. (36) and (37), we obtain

r′L =|vR|∆t

2 sin(nα)= rL

θ

2 sin(nα)(38)

11

where rL = |vR|/ωc is a physical Larmor radius. For n = 1, we obtain

a familiar result of r′L = rL[1 + (θ/2)2]1/2 from Eq. (7). In general cases,

Eq. (38) can be expanded to

r′L = rL

(1 +

[ 1

24+

1

12n2

]θ2 +

7n4 − 20n2 − 32

5760n4θ4 +O(θ6)

)(39)

Figure 2(b) shows the relative length of Larmor radius, r′L/rL = θ/[2 sin(nα)].

In the square brackets in Eq. (39), the first term is insensitive to n, while the

second depends on n. Thus, higher n does not allow n-times larger timestep.

To keep an error within ≈ 1%, we need to limit the timestep to

θ = ωc∆t . 0.5

√n2

n2 + 2. (40)

For n → ∞, we find ωc∆t < 0.5. If we set the threshold to ε%, both

Eqs. (35) and (40) can be relaxed by a factor of√ε. For example, Eq. (40)

gives ωc∆t < 1 for 4%.

The multiple Boris solvers have two nice properties for stable solvers.

The first one is the time symmetry. As evident in Eqs. (3), (18), and (5), the

multiple Boris solvers is symmetric in time. In such a case, it is unlikely that

the solvers numerically damp or amplify a motion in one time direction. In

Eq. (20), procedures in the reverse-time direction are obtained by reversing

the sign of the second coefficient, i.e., c−n1 = cn1, c−n2 = −cn2, and c−n3 =

cn3. It is obvious that c01 = 1, c02 = 0, and c03 = 0 for n = 0. As a result, we

have the coefficients not only for positive integers, but also for any integer n.

The second is the volume preservation [4, 5], which is related to the long-

term stability. We consider the evolution of a volume in the phase-space from

t − ∆t2

to t+ ∆t2

. Since the Lorentz-force part of the multiple Boris solver

is equivalent to n-times repetition of the Boris-type element, we express the

12

equivalent numerical procedures into the following steps [5] of S1, S2, · · ·S5,

where Sk indicates the k-th step. The third step, the Lorentz-force part of

the solver, is further divided into n substeps. Boris(u,∆t) indicates the

Boris-type procedure (Eqs. (4a)–(4c)) on the vector u for a timestep ∆t.

S1 : xt = xt−∆t2 +

∆t

2

ut−∆t2

γt−∆t2

, ut−∆t2 = ut−∆t

2 (41)

S2 : xt = xt, u− = ut−∆t2 + ε∆t (42)

S31 : xt = xt, u+1 = Boris(u−,

∆t

n) (43)

...

S3m : xt = xt, u+m = Boris(u+

m−1,∆t

n) (44)

...

S3n : xt = xt, u+n = Boris(u+

n−1,∆t

n) (45)

S4 : xt = xt, ut+ ∆t2 = u+

n + ε∆t (46)

S5 : xt+ ∆t2 = xt +

∆t

2

ut+ ∆t2

γt+∆t2

, ut+ ∆t2 = ut+ ∆t

2 (47)

We check the Jacobian Jk for the k-th step, Sk.

Jk =∣∣∣∂(xnew,unew)

∂(xold,uold)

∣∣∣ = det

∂xnew

∂xold

∂xnew

∂uold

∂unew

∂xold

∂unew

∂uold

. (48)

For the first two and the last two steps (S1, S2, S4, and S5), we obtain

J1 = J5 =

∣∣∣∣∣∣I∂xnew

∂uold

0 I

∣∣∣∣∣∣ = 1, J2 = J4 =

∣∣∣∣∣∣ I 0

∂unew

∂xold I

∣∣∣∣∣∣ = 1 (49)

13

In addition, Zhang et al. [5] proved that the Lorentz-force part of the rel-

ativistic Boris solver (Boris(u,∆t)) preserves a phase-space volume,1 and

the proof holds true for an arbitrary timestep ∆t. Then, for the substeps in

the third step (S31, · · ·S3m, · · ·S3n), each Boris-type element (Boris(u, ∆tn

))

preserves the phase-space volume, i.e., J31 = · · · = J3m = · · · = J3n = 1.

As a result, the Jacobian throughout the entire procedure satisfies Jall =

J1J2(J31 · · · J3m · · · J3n)J4J5 = 1. This indicates that the entire procedure

preserves the phase-space volume. The multiple Boris solvers are volume-

preserving.

4. Numerical tests

To evaluate the new solvers, we have conducted four numerical tests. For

comparison, the following ten particle integrators are benchmarked:

1. Classical Boris solver (Eqs. (4a)–(4c)),

2. Single Boris solver (n = 1; Eqs. (20), (30))

3. Double Boris solver (n = 2; Eqs. (20), (31))

4. Quadruple Boris solver (n = 4; Eqs. (20), (33)).

5. Octuple Boris solver (n = 8; Eqs. (20), (22)–(25)).

6. 16x Boris solver (n = 16; Eqs. (20), (22)–(25)).

7. 32x Boris solver (n = 32; Eqs. (20), (22)–(25)).

8. Umeda solver (N = 3) [6]

9. Umeda solver (N = 4) [7]

1The authors claimed that they developed a new volume-preserving algorithm, but the

algorithm appears to be the standard relativistic Boris solver.

14

10. Exact-gyration solver [8]

The first one is the classical Boris solver [1, 2]. The next six are the

multiple Boris solvers with n = 1, n = 2, n = 4, n = 8, n = 16, and n = 32.

The coefficients are given by Eqs. (30)–(33) or Eqs. (22)–(25). We refer to

them as the single, double, quadruple, octuple, 16x and 32x Boris solvers,

respectively. The next two are the Umeda solvers with N = 3 steps [6] and

N = 4 steps [7]. We follow procedures in Eqs. (8)–(10) in Ref. [7]. As they

contain 2N−2 Boris elements, they are equivalent to the double and quadruple

Boris solvers. The last one is Zenitani & Umeda [8]’s solver, which directly

calculates Euler’s rotation formula in the Lorentz-force part:

θ ≡ qB

mγ∆t, (50)

u‖ = (u · b̂) b̂ (51)

u+ = u‖ + (u− u‖) cos θ + (u× b̂) sin θ (52)

By definition, this solver does not produce an error during the gyration phase.

We refer to it as the exact-gyration solver. Since it uses trigonometric func-

tions, its computation cost is usually higher than those of polynomial-based

solvers. Note that all these solvers share the same Coulomb-force procedures

(Eqs: (3) and (5)).

In the first test, we have run test-particle simulations in a static magnetic

field B = (0, 0, 1) and E = (0, 0, 0). Parameters are set to m = 1, q = 1,

and c = 1. The initial condition is u(t = 0) = (1, 0, 0) and the timestep is

set to ∆t = π/60, π/20, π/6, or π/2. We have calculated errors in u from

the analytic solutions. We have run the simulations until t = 12π, and then

we have evaluated the maximum errors in a 4-vector, {|δu|/|u|}max. Since

15

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k�ukkuk max

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Figure 3: Normalized errors in u by the test-particle simulations as a function of θ = ωc∆t.

16

ωc = qB/(mγ) = 1/√

2, the interval 12π corresponds to ≈ 4 rotations. This

test is a subset of numerical tests in our recent study [8]. Here, almost all of

the solvers are combinations of the exact solver for the Coulomb-force parts

(Eqs. (3) and (5)) and the second-order solver for the Lorentz-force part.

These solvers only differ in the Lorentz-force part.

Figure 3 shows the results as a function of ∆t. Errors by the classical

Boris solver and the Umeda solvers are not presented, because the errors are

identical to those by the single, double, and quadruple Boris solvers. One

can see that the multiple Boris schemes give second-order errors with respect

to ∆t. The errors also scale like ∝ n−2. This is because the error simply

accumulates in the interval of our interest. From Eq. (34), the errors can be

estimated to δθ ≈ (ωc12π) 112

(ωc∆t/n)2 = (π/√

2)(ωc∆t/n)2. The results are

in excellent agreement with the prediction. Meanwhile, the exact-gyration

solver only gives small errors of O(10−13) ∼ O(10−14).

In the second test, we have measured elapse times of test-particle simu-

lations. Since algorithms’ performance relies on CPU architectures, we have

run the code on Intel and SPARC processors. We have compiled the pro-

gram on an Intel Core i7 processor with Intel Fortran compiler 2019. For each

solver, we have run the program for 1.2× 107 timesteps. We have measured

the elapse time six times for every cases. The black bars in Figure 4 show the

results, in normalized units. We have also benchmarked the program on a

SPARC processor with FUJITSU Fortran compiler on FX100 supercomputer

at the Japan Aerospace Exploration Agency (JAXA). The elapsed times are

averaged over three runs, because the results on the SPARC processor have

less variations than those on the Intel processor. The results are presented

17

0.50 0.75 1.00 1.25 1.50

Classical

Single (n=1)

Double (n=2)

Quadruple (n=4)

Octuple (n=8)

16x (n=16)

32x (n=32)

Umeda (N=3)

Umeda (N=4)

Exact-gyration

5.06518.21

5.32515.77

5.57915.81

5.70515.83

5.89917.79

6.38718.76

6.68619.47

7.19319.26

7.8820.35

6.44822.65

IntelSPARC

Figure 4: Benchmark results of the test-particle simulations. The average elapse times

(in seconds) are normalized to those of the classical Boris solver.

18

in the gray bars in Figure 4. To emphasize the difference, the leftmost value

is set to 0.5. Note that we show total elapse time of test-particle simula-

tions. The programs contain not only the Lorentz-force part, but also the

Coulomb-force parts (Eqs. (3) and (5)) and the position part (Eq. (1)). Thus

the results are not proportional to, but highly affected by the computation

time of the Lorentz-force solvers.

As far as we have tested, the new solvers are adequately fast. On the Intel

processor, they are slower than the classical Boris solver, but faster than the

Umeda solvers and the exact-gyration solver except for the n = 32 case.

The multiple Boris solvers become slower as n increases. On the SPARC

processor, the new solvers are more promising than on the Intel processor.

Some of them (n = 1–4) are 50% faster than the exact-gyration solver and

even faster than the classical Boris solver. It is noteworthy that the double

Boris solver (n = 2) is significantly faster than the Umeda solver (N = 3),

although the two solvers are equivalent. Similarly, the quadruple Boris solver

(n = 4) is faster than the Umeda solver (N = 4). Surprisingly, even the 32x

Boris solver (n = 32) looks as good as the Umeda solver (N = 3), even

though it gives 256 times higher accuracy.

In the third test, we have run test-particle simulations with the single

(n = 1) Boris solver, the quadruple (n = 4) Boris solver, and the exact-

gyration solver in various configurations. We consider five configurations,

as shown in Table 1. In cases 1 and 2, the electric field E is dominant:

|E| � c|B|. In case 3, the electric and magnetic fields are equal: |E| =

c|B|. In cases 4 and 5, the magnetic field B is dominant: |E| � c|B|. In

particular, case 5 stands for the gyration about B, as studied in the first

19

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k�ukkuk max

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Figure 5: Normalized errors in u in the test-particle simulations, as a function of θ =

ωc∆t. The single (n = 1) Boris solver (dotted lines), the quadruple (n = 4) solver (solid

lines) and the exact-gyration solver (dashdotted lines) are tested in five configurations

(colors; Table 1).

20

# B E

1 (0,0,0) (1,0,0) direct acceleration by E

2 (0,0,0.1) (1,0,0) E-dominated

3 (0,0,1) (1,0,0) |E| = c|B|4 (0,0,1) (0.1,0,0) E×B drift

5 (0,0,1) (0,0,0) gyration about B

Table 1: Field settings for test-particle simulations.

test. Other parameters, timesteps, and the initial conditions are similar to

those in the first test. We evaluate maximum values of the normalized error

(|δu|/|u|)max during the time interval of 0 < t < 12π. Here, δu is deviations

from analytic solutions (cases 1 and 5) or numerical reference values (cases

2–4). We obtain the numerical reference values by using the exact-gyration

solver with ∆t = π/240. These configurations and procedures are the same

as in Ref. [8].

Figure 5 shows the errors as a function of the timestep. The dotted, solid,

and dashdotted lines indicate the results by the single (n = 1) Boris solver,

the quadruple (n = 4) solver, and the exact-gyration solver, respectively. In

case 1 (in red) by the three solvers and case 5 by the exact-gyration solver,

one can essentially see no error. All the other curves suggest the second-

order accuracy. In cases 2 and 3, the three solvers return similar results.

Meanwhile, in cases 4 and 5, their results are different. The quadruple Boris

solver provides 10.9–16.5 times smaller errors than the single Boris solver,

as emphasized by the solid arrows in green and blue. The improvement is

asymptotic to 16 for ωc∆t� 1 in case 5. The exact-gyration solver provides

21

even better results, as indicated by the dashed arrows. The quadruple Boris

solver provides similar errors in cases 4 and 5. So does the single Boris solver.

It appears that the accuracy in the Lorentz-force solver (case 5) limits the

overall accuracy for |E| � c|B| (case 4).

In the fourth test, we study the long-term stability of the particle motion

in a non-uniform electromagnetic field. As done in Refs. [4, 8], we have run

test-particles in the following field,

B = (x2 + y2)1/2ez, φ = 0.01(x2 + y2)−1/2. (53)

The conditions are set to u(t = 0) = (0.1, 0.0), x(t = 0) = (0.9, 0, 0),

m = c = 1, and ∆t = π/10. We compare the single (n = 1) Boris solver,

the quadruple (n = 4) Boris solver, the exact-gyration solver, and the fourth

order Runge–Kutta solver. The blue and gray lines in Figure 6 show the

trajectories by the quadruple Boris solver and by the Runge–Kutta solver,

at (a) an initial stage and at (b) a late stage. The time history of the total

energy is presented in Figure 6(c). It combines the particle kinetic energy

and the electric potential energy at the midpoint positions (x(t+ ∆t2

)). The

values are normalized by the initial values.

One can see typical trajectories in Figure 6(a). It consists of a fast small-

scale gyration and a slow large-scale rotation. The large-scale rotation is

due to the ∇B drift and the E×B drift. In the case of the quadruple Boris

solver, the particle keeps gyrating even after a long time (300th turn; t ≈2× 105) in Fig. 6(b). The particle energy is excellently conserved. We only

recognize a relative error of O(10−4). The trajectories (not shown), the time

histories, and the error amplitudes (not shown) by the single (n = 1) Boris

solver and the exact-gyration solver are similar to those of the quadruple

22

(a) 1st turn (b) 300th turn

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RK4

Quadruple Boris

Quadruple Boris

(c) Time history 300th turn

Figure 6: Particle trajectories by the quadruple Boris solver at (a) the initial stage and

(b) the late stage. A trajectory by the RK4 solver is overplotted in gray at (b) the late

stage. (c) Energy histories by the four solvers.

23

(n = 4) Boris solver. In contrast, as evident in the gray line in Fig. 6(b), the

particle gyroradius becomes smaller and smaller in the case of the Runge–

Kutta solver. The particle numerically lost 30% of its total energy (Fig. 6(c)).

These results are attributed to the volume-preserving property of the

numerical schemes. It is known that the Runge–Kutta solver is not volume

preserving. The solver is incapable of preserving the energy of the small-scale

oscillation. On the other hand, as discussed in Section 3, the multiple Boris

solvers are volume-preserving. The other two solvers are also known to be

volume-preserving [4, 5, 8]. These three schemes are capable of preserving

the energy of the small-scale oscillation, even in a nonuniform field.

5. Discussion and Summary

We have proposed the multiple Boris integrators that virtually use the

Boris procedure n times in the Lorentz-force part. The schemes provide

second-order but n2-times smaller errors (Eq. (34)) than the classical Boris

solver. The coefficients are given by either the generic forms of p (Eqs. (23)–

(25)), the generic forms of t (Eqs. (27)–(29)), or the explicit forms (Eqs. (30)–

(33)). In our experience, the generic forms with p are useful for large n.

For small n (n ≤ 4), we prefer the explicit form, because we can further

optimize the code. In our implementation for n = 1 . . . 4, we have calculated

the numerators and denominators in Eqs. (30)–(33) separately, because the

explicit forms share the same denominator, (1 + t2)n. As t contains 1/γ

(Eq. (17)), we multiply the numerators and denominators by γ2n. By doing

so, we reduce the number of division operations to save the total computation

cost.

24

The new solvers are developed in the same spirit as in the multistep

Boris solvers [6, 7], which considered 2N−2-times smaller timesteps in the

Lorentz-force part. In the multiple Boris solvers, we virtually divide the

rotation angle into n Boris units. This n is arbitrary and not limited to

2N−2. Thus one can choose the best n for one’s application. In addition,

we calculate the vector equation (Eq. (20)) only once, because we prepare

the coefficients in advance. The coefficients are rational fractions, and so

it is easy to calculate them. Consequently, the program runs efficiently, as

evident in the benchmark results.

We have also evaluated numerical accuracy of several solvers in general

cases of B 6= 0 and E 6= 0. Figure 5 demonstrates the second-order accuracy

in most cases, because the second-order Lorentz-force solvers are combined

with the Coulomb-force procedures (Eqs. (3) and (5)) in an operator-splitting

manner. In this study, since all the Lorentz-force solvers are combined with

the same Coulomb-force procedures, the numerical accuracy for |E| � c|B| isessentially controlled by the Lorentz-force solvers, even though the resulting

schemes provide the same second-order accuracy. This is evident in cases

4 and 5 in Figure 5 — the quadruple (n = 4) Boris solver provides ≈ 16

times better accuracy, and the exact-gyration solver, which corresponds to

the multiple Boris solver in the n→∞ limit, provides even better accuracy.

In general, the multiple Boris solvers have higher accuracy than the classic

Boris solver, and then they bring better accuracy in the |E| � c|B| cases.

Importantly, we often consider |E| � c|B| in practical applications in PIC

simulation.

The new solvers may further reduce the computation cost, by allowing

25

larger timesteps. In PIC simulation, the timestep ∆t is limited by the plasma

frequency ωp and the gyrofrequency ωc. The ratio of ωp to ωc depends on

applications, and both frequencies are limited to ω∆t . O(0.1) due to the

numerical accuracy and ω∆t < 2 due to the numerical stability [1]. If the

gyrofrequency limits the timestep, the multiple Boris solvers can relax the

limitations for the accuracy. As evaluated in Eq. (35), the n-tuple Boris

solver allows n-times larger timestep, because it repeats the classical Boris

solver with a smaller timestep (∆t/n) in the Lorentz-force part. Actually,

in the other parts of PIC simulation, the timesteps are unchanged. For

example, we update the position (Eq. (1)) for ∆t, not for ∆t/n. This fact

makes the timestep condition more complicated. The new condition is given

by Eq. (40). This is more restrictive than Eq. (35).

In the nonuniform electromagnetic field, the multiple Boris solvers have

a favorable property of the long-term stability. Since they inherit a volume-

preservation property from the Boris-type element, the new solvers are ca-

pable of preserving a small-scale oscillation, as they should be. As far as

we have tested, they conserve the total energy very well. These facts give

us further confidence to use the new solvers in PIC simulations for practical

applications.

Among the new solvers, considering the accuracy, the timestep, and com-

putation costs, we recommend the quadruple (n = 4) Boris solver with

ωc∆t . 0.5 to the readers. One can develop higher-order solvers (n � 32)

to obtain even better accuracy. In such a case, we recommend the exact-

gyration solver [8] to the readers. As shown in Figure 4, the 32x Boris solver

(n = 32) is already more expensive than the exact-gyration solver on In-

26

tel processors. It is likely that higher-order solvers are as expensive as the

exact-gyration solver on SPARC processors for n � 32. In any case, since

the extra cost is limited, the exact-gyration solver will be another choice for

n > 10. Finally, we note that our recommendation is partly based on the

benchmarks on our specific systems. Performance may be different on other

systems such as GPU-based systems and FGPA systems. In general, since it

is impossible to predict future computing platforms, we can only prepare as

many particle solvers as possible. Our solvers will be the latest additions.

In summary, we have proposed the multiple Boris solvers that virtually

combine multiple Boris units in the Lorentz-force part. We have derived one-

step expressions with polynomial-based coefficients for arbitrary n divisions.

The new solvers provide second-order, n2-times smaller errors than the clas-

sical Boris solver. They also allow larger timesteps. Their performance is

promising. They are stable over a long time. We hope that the proposed

solvers will be useful in studies with PIC simulations.

Acknowledgements

This work was partially carried out on facilities of the JSS2 system at

Japan Aerospace Exploration Agency (JAXA). This work was supported by

Grant-in-Aid for Scientific Research (C) 17K05673, (S) 17H06140, and (B)

17H02877 from the Japan Society for the Promotion of Science (JSPS).

27

Appendix A. Third coefficients for n = 2k + 1

We recall the following relation of trigonometric functions.

sin(A+ α) + sin(A− α) = 2 sinA cosα (A.1)

We substitute (2k + 1)α to A, where k is a positive integer, k ≥ 1. Using

Chebyshev polynomials, we obtain

2 sin((2k + 1)α

)cosα = sin

(2(k + 1)α

)+ sin

(2kα

)= sin 2α

(Uk(cos 2α) + Uk−1(cos 2α)

)(A.2)

sin((2k + 1)α

)= sinα

(Uk(cos 2α) + Uk−1(cos 2α)

)(A.3)

With help from Eqs. (7), (8), and (22), we obtain

cn3 =2

t2sin2

((2k + 1)α

)=

2

1 + t2

{Uk

(1− t21 + t2

)+ Uk−1

(1− t21 + t2

)}2

= (1 + p){Uk(p) + Uk−1(p)

}2

(A.4)

28

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[3] J. P. Boris, in Proceedings of 4th Conference on Numerical Simulation of

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20 (2013) 084503.

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[8] S. Zenitani, T. Umeda, Phys. Plasmas 25 (2018) 112110.

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29