7th Asia-Pacific Conference on Plasma Physics, 12-17 Nov, 2023 at Port Messe Nagoya

Recent advances in numerical schemes for plasma particle-in-cell simulations
Takayuki Umeda1, Harune Sekido1, Riku Ozaki1
1Institute for Space and Earth Environmental Research, Nagoya University
e-mail (speaker): umeda@isee.nagoya-u.ac.jp

The particle-in-cell method is first established forplasma simulations, and has now become widely used invarious scientific fields. A plasma particle-in-cellsimulation code consists of three kernels. The first is theMaxwell equations for electromagnetic fields, the second
is the equation of motions for charged particles, and thethird is the continuity equation for charge. In this paper,
recent advances in numerical schemes for these kernels
are briefly introduced.
The Finite-Difference Time-Domain (FDTD) method[1] has been used as the standard numerical method forsolving the Maxwell equations for over a half century. Its
4th-order version is known as “FDTD(2,4)” [2].
Although a higher-order finite difference reduces anumerical error in phase velocity, the Courant condition
is more restricted. Recently a new scheme is developed
for relaxing the Courant condition of FDTD(2,4) [3]. The
implementation of the new FDTD method into aparticle-in-cell simulation is discussed in the presentstudy.
Numerical methods for solving the relativistic motionof charged particles with a higher accuracy is an issue forcomputational physics in various fields. The classic
fourth-order Runge-Kutta method (RK4) [4] has beenused over many years for tracking charged particle
motions, although RK4 does not satisfy any conservation
law. However, the Boris method [5] has been used over ahalf century in particle-in-cell plasma simulations
because of its property of the energy conservation during
the gyro motion. Recently, a new method for solving
relativistic charged particle motions has been developed[6], which conserves both boosted Lorentz factor andkinetic energy during the gyro motion. However, the new
integrator has the second-order accuracy in time and isless accurate than RK4. In the present study, a new
fourth-order method has been developed by combining
the new integrator [6] and RK4. The proposed method
conserves boosted Lorentz factor and kinetic energy as

well during the gyro motion, but has higher accuracy
than RK4.
It has been known that current densities without thecontinuity equation for charge includes a numerical error
in electrostatic fields, which needs to be corrected bysolving the Poisson equation implicitly. The charge
conservation scheme for computing current densities has
been discussed by many authors since early 1990’s[7-11], which provides current densities that exactly
satisfies the continuity equation for charge. Extension ofthe charge conservation scheme to higher-order particle
shape function is discussed in the present study.
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Propag., 71, 1630—1639 (2023)[4] J. C. Butcher, Appl. Num. Math., 20, 247—260(1996)[5] J. P. Boris, in Proceedings of 4th Conference on
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