Boris-type particle integrators inparticle-in-cell (PIC) simulation
Seiji ZENITANIRCUSS, Kobe University

Tsunehiko Kato
National Astronomical Observatory of Japan

Takayuki Umeda
Nagoya University

Particle-in-cell (PIC) simulation

	xn → xn +1
vn +1/ 2 → vn +3 / 2
Bn , E n

J n +1/ 2 , ρ n

Bn → Bn +1
En → En +1

Accurate modeling ofparticle motion isimportant for studying
kinetic plasma processes(reconnection, shocks,
kinetic turbulence…)

Particle solver• Particle solver

u= v<latexit sha1_base64="2pCK63GPyGKcW6mbeYbgMQrqXns=">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</latexit>• Time-splitting
1/2-acceleration by E
gyration about B
1/2-acceleration by E

Boris solver (2-step Boris solver)✓=✓<latexit sha1_base64="6Tuqpc0iHvQertO7kAS733vJda4=">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</latexit>

⇣1

12

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+
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arctan(✓/2)
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Boris 1970
Hockney & Eastwood 1981
Birdsall & Langdon 1985

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2<latexit sha1_base64="7QaLj6wOtgrEGlbeWVRoxBmdfXw=">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</latexit>

2
2
⇥t
u′0 ×
2
1+t

Can we eliminate this error?

(Solution 0: tan correction by [Boris 1970])

Solution 1: From a different angle…
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sha1_base64="QJQISasCOAMHt7xpa1LuFoZYNMw=">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</latexit>
sha1_base64="wnZw/rosDQm35RL9qdz2hAzSmKk=">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</latexit>
sha1_base64="OKklI08EYzNlafZuysMNFNcOjz8=">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</latexit>

u+
exact
u

u0

+

u+
exact

u00

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sha1_base64="DsvVYubsmZ1qdvPrgZNNLDIdXMc=">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</latexit>

↵

u ⇥t

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↵

u0

2↵

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2↵u<latexit sha1_base64="f8EiUDlQUkU+gTQrbHNZblOkTlY=">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</latexit><latexit sha1_base64="VIYPSaz7wNpUnqsjyMurLfgzuMM=">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</latexit><latexit sha1_base64="VIYPSaz7wNpUnqsjyMurLfgzuMM=">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</latexit>(a) Boris solveru ⇥t<latexit sha1_base64="RiOvcG/VLv+B7gIfQzQPoLeXe2M=">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</latexit><latexit sha1_base64="NWXifH6DoUOOs0RnpAg+13YzrjI=">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</latexit><latexit<latexit sha1_base64="nMYYCCnFB0kJOAQez4b6EVfIOEs=">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</latexit><latexit(b) Double Boris solver

Rotation by an approximate angle 2αerSolution 1: Multiple Boris solvers<latexit sha1_base64="e3MAIc11EBXuIPnEG2FS4PHfo8Y=">AAADY3ichVLLTttAFD2Jobza8lwgoUpRI6SuonGFxBaVTZcgCCCRCI2dIVj4VXsShCL/QH+AbVkifoQfYMEHsGDNgg0LjifOAqI0Y9m+j3PuvXPvdWLfS7UQD6WyNTH5aWp6Znbu85ev8wuLSwdp1ElcVXcjP0qOHJkq3wtVXXvaV0dxomTg+OrQOd/O/YddlaReFO7ry1g1A9kOvVPPlZqmZqOlfC0rDX2mtDxZqIqaMKcyLNiFUEVxdqLFUgUNtBDBRQcBFEJoyj4kUj7HsCEQ09ZEj7aEkmf8Chlmye0QpYiQtJ7z26Z2XFhD6nnM1LBdZvH5JmRWsC7uxY14FnfiVjyJ15GxeiZGXssl/06fq+KT+b+rey9kjeY5xGdYJ94xmVuUAno1zsZmG+D+x0/MDRUuzM0C4wuJ6tHXpd0lO+9f01gGMQb3yLUeqvRnRSdHxWozk6ScGS3vqWPmk42tosVZdYd4+cRdM3M9JnNSYN7zG2YH+nMlnztnf9ywYeHgZ80WNXt3o7r1q9i+aazhO36wA5vYwm/soM4a/uAK/3BdfrTmrCVrpQ8tlwrOMt4d69sb04K0dQ==</latexit>

✓

u

+

n=2

<latexit sha1_base64="DsvVYubsmZ1qdvPrgZNNLDIdXMc=">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</latexit>
sha1_base64="BHd3Sl7aP2BxNKkRIzzoC0XyaoQ=">AAADSHichVLJTsJQFD2AIuIA7EzcEImJK/Nw49bEjUuMMiRKSFue2NApbcEQ0h9g61cYf8QfMMZPcO3CjSaePsoCCfKatucO5953B92zzCAU4j2VzqytZzdym/mt7fzObqG43QjcgW/IuuFart/StUBapiProRlasuX5UrN1Szb1/nlsbw6lH5iucx2OPNm2tZ5j3pmGFlJV6xQr4lioU14E1QRUkBy3lCrjFl24MDCADQkHIbEFDQGfG1Qh4FHXxpg6n8hUdokIeXIH9JL00Kjt89ujdJNoHcpxzECxDWax+PpklnEoXsWT+BQv4ll8iO+lscYqRnyXEf/6lCu9TmGyd/VF1nKeTv8Ih/TXVeYukU1riPuV2WZ+//F9VaHEg6rMVjaHXmPahtQbZMf9ayvNLMasjlgao0J7lHRyWaweM2nEkZLinupqPtHKW3Q5q+ECL564oWYersjsJz7z/Fu1A9O5ks+Vq/5dsEXQODmuEl8K5LCPAxyx8FOc4QI11NV1JnhMv6V/MmvT1Uynkh0tYe5ksr8hmK5H</latexit>
sha1_base64="QJQISasCOAMHt7xpa1LuFoZYNMw=">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</latexit>
sha1_base64="wnZw/rosDQm35RL9qdz2hAzSmKk=">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</latexit>
sha1_base64="OKklI08EYzNlafZuysMNFNcOjz8=">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</latexit>

u+
exact

u00

<latexit sha1_base64="sr0WCmcpZ2nM8+P2OMwqm+cnoZo=">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</latexit>u<latexit sha1_base64="2D/uO79zanxHoZtPQ1qM2buUv7E=">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</latexit>

0

u ⇥t

• We repeat Boris procedure
multiple times (Inspired by
Umeda 2018)<latexit sha1_base64="GlPJeIMadP0khNZ5/LSf+Ehz1aE=">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</latexit>

u02↵<latexit sha1_base64="RiOvcG/VLv+B7gIfQzQPoLeXe2M=">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</latexit><latexit sha1_base64="2D/uO79zanxHoZtPQ1qM2buUv7E=">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</latexit><latexit sha1_base64="f8EiUDlQUkU+gTQrbHNZblOkTlY=">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</latexit>✓u ⇥t<latexit sha1_base64="RiOvcG/VLv+B7gIfQzQPoLeXe2M=">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</latexit><latexit sha1_base64="NWXifH6DoUOOs0RnpAg+13YzrjI=">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</latexit><latexit↵
u<latexit sha1_base64="N0dOw62fRJxjDChiNjS8eCPk2KU=">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</latexit><latexit

2↵<latexit sha1_base64="f8EiUDlQUkU+gTQrbHNZblOkTlY=">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</latexit>✓⌘<latexit sha1_base64="VIYPSaz7wNpUnqsjyMurLfgzuMM=">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</latexit><latexit sha1_base64="JqHPrQfncR/IJVIs8XbuJCYdZA0=">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</latexit>

q t
B
m

(b) Double Boris solver--

Zenitani & Kato 2019

Solution 1: Multiple Boris solvers

-

-

-

General form for an arbitrary nt⌘<latexit sha1_base64="j6OD1BHb0S3nVf5+WFp9i86i+Z0=">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</latexit>

qB t
2nm

cn1 = Tn (p)cn2 = (1 + p)Un 1 (p)
8
1+p
>
>
⇣
⌘2
<
(1 + p) Uk (p) + Uk 1 (p)
cn3 =
⇣
⌘2
>
>
: 2 (1 + p)U(p)
k 1(for n = 1)
(for n = 2k + 1)
(for n = 2k)<latexit sha1_base64="MasLUimsYuUNJLzJC8ql1n5Yi2A=">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</latexit>

Zenitani & Kato 2019

Reminder: Chebyshev polynomials

https://en.wikipedia.org/wiki/Chebyshev_polynomials#Examples

Solution 2: Exact-gyration solver(u · B) B
uk =
|B|2
+

u = uk + (u<latexit sha1_base64="B3hd2dyej+1CKLOUGMcbbKvHPWY=">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</latexit>
sha1_base64="rkI7kiVJSWO2j/urgI31V94UHw4=">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</latexit>
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• Rotation angle

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sha1_base64="rkI7kiVJSWO2j/urgI31V94UHw4=">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</latexit>
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sha1_base64="LNnrDpjNZgDRZ73tML1+HPUDojc=">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</latexit>

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arctan(✓/2)<latexit sha1_base64="+2kx4WfuydkMzLQeGZxr6WQShW8=">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</latexit>• Equivalent to the Boris
solver with a gyrophase correctionu<latexit sha1_base64="VIYPSaz7wNpUnqsjyMurLfgzuMM=">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</latexit>

Zenitani & Umeda 2018

Stability: Volume preservation• Stable scheme preserves a volume in the phase space
during the temporal evolution (Qin et al. 2013).
• We have formally proved this property in our paper.

(x

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t

t

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Stability: Volume preservation
2-step Boris: volume-preserving

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RK4

Qin et al. 2013

RK4

B

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Constraint to the timestep• Because of a numerical Larmor radius,{ n2/(2+n2) }0.5 times larger Δt is allowed

Numerical
gyroradiusωcΔtZenitani & Kato 2019

Numerical tests (1): Errors in gyration⇣ t ⌘2
/
n

Error

k uk

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Zenitani & Kato 2019

Numerical tests (2): Errors in various field• Errors are largely controlled by
the gyration part, because all
the solvers share the same
Coulomb-force solver

Error

E
E

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kuk max

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Zenitani & Kato 2019

Numerical tests (3): Computation timeMultiple
Boris

Exactgyration
Fast

Slow

Zenitani & Kato 2019

Summary• Multiple Boris solvers• n-times multiplication of the Boris solver• Single-step formula with Chebyshev polynomials• n2-times higher accuracy for the gyration part• Exact-gyration solver• Based on the rotation formula• Exact accuracy for the gyration part

• Both solvers are computationally affordable• References:• Zenitani & Umeda, On the Boris solver in particle-in-cell simulation, Phys.
Plasmas. 25, 112110 (2018)• Zenitani & Kato, Multiple Boris integrators for particle-in-cell simulation,
Comput. Phys. Commun., in press (2019)