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Boris-type particle integrators in particle-in-cell (PIC) simulation Seiji ZENITANI RCUSS, Kobe University Tsunehiko Kato National Astronomical Observatory of Japan Takayuki Umeda Nagoya University

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Page 1: Boris-type particle integrators in particle-in-cell (PIC) simulationBoris-type particle integrators in particle-in-cell (PIC) simulation Seiji ZENITANI RCUSS, Kobe University Tsunehiko

Boris-type particle integrators in particle-in-cell (PIC) simulation

Seiji ZENITANI RCUSS, Kobe University

Tsunehiko Kato National Astronomical Observatory of Japan

Takayuki Umeda Nagoya University

Page 2: Boris-type particle integrators in particle-in-cell (PIC) simulationBoris-type particle integrators in particle-in-cell (PIC) simulation Seiji ZENITANI RCUSS, Kobe University Tsunehiko

Particle-in-cell (PIC) simulation

2/32/1

1

++

+

nn

nn

vvxx

1

1

+

+

nn

nn

EEBB

nn EB ,nnJ ρ,2/1+

Accurate modeling of particle motion is important for studying kinetic plasma processes (reconnection, shocks, kinetic turbulence…)

Page 3: Boris-type particle integrators in particle-in-cell (PIC) simulationBoris-type particle integrators in particle-in-cell (PIC) simulation Seiji ZENITANI RCUSS, Kobe University Tsunehiko

• Particle solver

Particle solver

• Time-splitting

1/2-acceleration by E

gyration about B

1/2-acceleration by E

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Page 4: Boris-type particle integrators in particle-in-cell (PIC) simulationBoris-type particle integrators in particle-in-cell (PIC) simulation Seiji ZENITANI RCUSS, Kobe University Tsunehiko

• Second-order accuracy in angle

u0Boris

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u+Boris

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u+New

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arctan(✓/2)<latexit sha1_base64="2k7Jkkcp1ZHrhp1lSLPmSRlf5SU=">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</latexit><latexit sha1_base64="2k7Jkkcp1ZHrhp1lSLPmSRlf5SU=">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</latexit><latexit sha1_base64="2k7Jkkcp1ZHrhp1lSLPmSRlf5SU=">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</latexit><latexit sha1_base64="2k7Jkkcp1ZHrhp1lSLPmSRlf5SU=">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</latexit>

u� ⇥ ✓

2<latexit sha1_base64="XtQ5EFXzYybfw4qF8iuBDD6BBUg=">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</latexit><latexit sha1_base64="XtQ5EFXzYybfw4qF8iuBDD6BBUg=">AAADf3ichVJNb9NAEB3XFEqhNIULEheLqGovjdZRpYpbVS4ci2jaSnGI1pttatVfWm9SVdb+ASR+SNUjV7jzBzjwEzhz6IUDzxvnEKKQtWzPvnlvZndmwjyOCs3YT2fFfbD68NHa4/UnTzeebTa2np8W2UgJ2RFZnKnzkBcyjlLZ0ZGO5XmuJE/CWJ6FV28r/9lYqiLK0hN9k8tewodpdBEJrgH1GzvBWIpyZD7ueYGOEll4wYXiorRwoC+l5saUbdNvNFmL2eXNG35tNKlex9mW41FAA8pI0IgSkpSShh0TpwJPl3xilAPrUQlMwYqsX5KhdWhHYEkwONArfIfYdWs0xb6KWVi1QJYYr4LSo232g92y3+w7u2O/2J+FsUobozrLDf7hRCvz/uanlx/uoVqsC8E3tA1+aDMPYCXwarpcmm3K+59e2RtKurY3S6wvBauEbwxcQF3Vr2eRaYzpPapdSU34TV3JRbGGyMRhG7urahra/pilpxigV+M5XdVxYXuul2RWNWdWH9gZmPQVesyc/++EzRun7ZbPWv77/ebhUT19a/SKXtMuKnBAh/SOjqmDM3ymL/SVvrmOu+O2XDahrji15gXNLPfNXxqcvyE=</latexit><latexit sha1_base64="XtQ5EFXzYybfw4qF8iuBDD6BBUg=">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</latexit><latexit sha1_base64="XtQ5EFXzYybfw4qF8iuBDD6BBUg=">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</latexit>

Boris solver (2-step Boris solver)

u−

u+

θ u′

2

1 + t2u′ × t

u− × t

Vay

1

2θ′

Boris 1970 Hockney & Eastwood 1981 Birdsall & Langdon 1985

�✓ = ✓⇣ 1

12✓2 � 1

80✓4 + · · ·

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2

1 + t2u0 ⇥ t

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2✓0

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Can we eliminate this error? (Solution 0: tan correction by [Boris 1970])

Page 5: Boris-type particle integrators in particle-in-cell (PIC) simulationBoris-type particle integrators in particle-in-cell (PIC) simulation Seiji ZENITANI RCUSS, Kobe University Tsunehiko

Solution 1: Multiple Boris solvers

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(a) Boris solver (b) Double Boris solver

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u+exact

<latexit sha1_base64="sr0WCmcpZ2nM8+P2OMwqm+cnoZo=">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</latexit><latexit sha1_base64="sr0WCmcpZ2nM8+P2OMwqm+cnoZo=">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</latexit><latexit sha1_base64="sr0WCmcpZ2nM8+P2OMwqm+cnoZo=">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</latexit><latexit sha1_base64="sr0WCmcpZ2nM8+P2OMwqm+cnoZo=">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</latexit>

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=">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</latexit><latexit sha1_base64="GlPJeIMadP0khNZ5/LSf+Ehz1aE=">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</latexit>

n=2

• We propose to repeat Boris procedure arbitrary n times

• We derive coefficientsfor a single-step procedure

u+n

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cn1 = Tn(p)

cn2 = (1 + p)Un�1(p)

cn3 =

8>><

>>:

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)

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Coefficients are give by

Page 6: Boris-type particle integrators in particle-in-cell (PIC) simulationBoris-type particle integrators in particle-in-cell (PIC) simulation Seiji ZENITANI RCUSS, Kobe University Tsunehiko

Reminder: Chebyshev polynomials

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

Page 7: Boris-type particle integrators in particle-in-cell (PIC) simulationBoris-type particle integrators in particle-in-cell (PIC) simulation Seiji ZENITANI RCUSS, Kobe University Tsunehiko

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u+B

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• Rotation angle

• Equivalent to the Boris solver with a gyro-phase correction

Solution 2: Exact-gyration solver

✓ =q�t

m��B<latexit sha1_base64="grMksTHKzqhsIB9OP+vuZvt3RTo=">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</latexit><latexit sha1_base64="grMksTHKzqhsIB9OP+vuZvt3RTo=">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</latexit><latexit sha1_base64="grMksTHKzqhsIB9OP+vuZvt3RTo=">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</latexit><latexit sha1_base64="grMksTHKzqhsIB9OP+vuZvt3RTo=">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</latexit>

u�k =

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Zenitani & Umeda 2018

Page 8: Boris-type particle integrators in particle-in-cell (PIC) simulationBoris-type particle integrators in particle-in-cell (PIC) simulation Seiji ZENITANI RCUSS, Kobe University Tsunehiko

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Numerical tests (1): Errors in gyration

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n

⌘2

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Zenitani & Kato 2020

Page 9: Boris-type particle integrators in particle-in-cell (PIC) simulationBoris-type particle integrators in particle-in-cell (PIC) simulation Seiji ZENITANI RCUSS, Kobe University Tsunehiko

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k�ukkuk max

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Numerical tests (2): Errors in various field

Error

Zenitani & Kato 2020

E

B

E×BE=BE

• Errors are largely controlled by the gyration part, because all the solvers share the same Coulomb-force solver

Page 10: Boris-type particle integrators in particle-in-cell (PIC) simulationBoris-type particle integrators in particle-in-cell (PIC) simulation Seiji ZENITANI RCUSS, Kobe University Tsunehiko

Numerical tests (3): Computation time

Fast Slow

Multiple Boris

Zenitani & Kato 2020

Exact-gyration

Page 11: Boris-type particle integrators in particle-in-cell (PIC) simulationBoris-type particle integrators in particle-in-cell (PIC) simulation Seiji ZENITANI RCUSS, Kobe University Tsunehiko

Constraint to the timestep

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

Zenitani & Kato 2020

Numerical gyroradius

ωcΔt

Page 12: Boris-type particle integrators in particle-in-cell (PIC) simulationBoris-type particle integrators in particle-in-cell (PIC) simulation Seiji ZENITANI RCUSS, Kobe University Tsunehiko

Stability: Volume preservation2-step Boris: volume-preserving

RK4

Exact/Zenitani: volume-preserving

Qin et al. 2013

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

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RK4

B

Page 13: Boris-type particle integrators in particle-in-cell (PIC) simulationBoris-type particle integrators in particle-in-cell (PIC) simulation Seiji ZENITANI RCUSS, Kobe University Tsunehiko

B Eγ=3

Ux0=2.83Uy=γVy

Ux=γVx

Ux=γVx

Uy=γV

y

Numerical boost in a magnetized flow (1/3)

• Conservation of force-free condition

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Page 14: Boris-type particle integrators in particle-in-cell (PIC) simulationBoris-type particle integrators in particle-in-cell (PIC) simulation Seiji ZENITANI RCUSS, Kobe University Tsunehiko

Second-order error

Boris-B

Zenita

ni

Numerical boost in a magnetized flow (2/3)

Page 15: Boris-type particle integrators in particle-in-cell (PIC) simulationBoris-type particle integrators in particle-in-cell (PIC) simulation Seiji ZENITANI RCUSS, Kobe University Tsunehiko

Lorentz factor γ

Force-free line

(Higuera=Cary)

Exact (Zenitani): A = 0

2-step Boris: A=-1/12

Numerical boost in a magnetized flow (3/3)

Over-gyration

Delay in gyration

Page 16: Boris-type particle integrators in particle-in-cell (PIC) simulationBoris-type particle integrators in particle-in-cell (PIC) simulation Seiji ZENITANI RCUSS, Kobe University Tsunehiko

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

• Numerical boost in a magnetized flow • Good solvers can reduce the numerical boost to 1/3

• 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. 247, 106954 (2020) https://doi.org/10.1016/j.cpc.2019.106954

• 銭谷誠司・加藤恒彦、相対論的プラズマ粒子シミュレーションのための粒子計算アルゴリズム、生存圏研究 14, 62 (2018) http://hdl.handle.net/2433/235378