24P149-P
Multi-Moment Advection Scheme for Vlasov Simulations
ブラソフシミュレーションのためのマルチモーメント移流法
Takashi Minoshima1 , Yosuke Matsumoto2 , Takanobu Amano3
簑島 敬1 , 松本 洋介 2 , 天野 孝伸 3
1

1

IFREE/JAMSTEC, 3173-25, Syowa-machi, Kanazawaku, Yokohama 236-0001, Japan
海洋研究開発機構 地球内部ダイナミクス領域 236-0001 神奈川県横浜市金沢区昭和町 3173-25
2
Department of Physics, Chiba University, 1-33, Yayoi-cho, Inage-ku, Chiba, 263-8522, Japan2
千葉大学理学研究科 263-8522 千葉県千葉市稲毛区弥生町 1-33
3Department of Physics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan3
名古屋大学理学研究科 464-8602 愛知県名古屋市千種区不老町

We present a new numerical scheme for solving the advection equation and its application to Vlasov simulations. The scheme treats not only point values of a profile but also its zeroth to second order piecewise
moments as dependent variables, and advances them on the basis of their governing equations. We show thatthe scheme provides quite accurate solutions within reasonable usage of computational resources compared toother existing schemes. Applications of the scheme to Vlasov simulations are presented with some benchmark
tests.

1 IntroductionThe Vlasov simulation, which directly discretizes theVlasov equation on grid point in phase space, hasbeen proposed for a collisionless plasma simulation
method, to remedy problems inherent to the common
Particle-In-Cell (PIC) simulation. A number of numerical schemes of the advection equation have been
proposed for the Vlasov simulation and succeeded inapplying to the electrostatic Vlasov-Poisson simulations thus far (e.g., [1]). However, the application to
the electromagnetic Vlasov simulation of magnetizedplasma is still limited, mainly owing to the difficulty
in solving the gyro motion around the magnetic field.
In this paper, we propose a new numerical scheme forthe advection equation, specifically designed to solve
the Vlasov equation in magnetized plasma.

∫∂M 0
∂
+ dx
(uf ) = 0,(3)
∂t
∂x
∫
∂M m1
∂
+(uxm f )
dx
∂tm!
∂x
∫
1
=uxm−1 f dx, (m = 1, 2) , (4)(m − 1)!
where u is the velocity. To solve a set of these equations, the scheme treats the point value of the profilefi and the piecewise moments as dependent variables,m
Mi+1/2=1m!

∫ xi+1

xm f dx, (m = 0, 1, 2) ,(5)xiand constructs a piecewise interpolation for f in a cell∑with a polynomial, Fi (x) = 5k=1 kCk;i (x − xi )k−1 .
The coefficients Ck;i are explicitly determined fromthe dependent variables at the upwind position as constraint. Then the variables are advanced on the basis of their governing equations (2)-(4) with the semi2 Multi-Moment Advection Scheme Lagrangian method. The multi-dimensional scheme isdesigned in a similar way. The scheme is termed as
The present scheme considers the advection of a prothe “Multi-Moment Advection (MMA)” scheme [2].
file f (x, t) and its zeroth to second order moments defined as∫1
3 Benchmark Testsm
xm f dx, (m = 0, 1, 2) .
(1)
M =m!
In one dimension, their governing equations are writ- Figure 1 shows the two-dimensional solid body rotation and advection problem of a symmetric gausten as
sian profile, solved by the MMA, CIP-CSL2 [3], and∂f∂+(uf ) = 0,(2) backsubstitution [4] schemes. While other schemes
∂t∂x

(b)

(c)

Vxe/U0(a)420-2-4

Vye/U0BKSB 84x84

420-2-4

Vxp/U00.4

CSL2 42x42210-1-2

Vyp/U0ωpeT = 31415.93, ωgeT = 314.16, mp/me = 25

MMA 34x342
10-1-20.0
1.0-0.2
50.0 rotations

50.0 rotations
0.5(d)

(e)

(f)

Y

0.2

0.0

0.0-0.2-0.4

300.0 rotations-0.4 -0.2 0.0 0.2 0.4
X

300.0 rotations-0.4 -0.2 0.0 0.2 0.4
X

300.0 rotations-0.4 -0.2 0.0 0.2 0.4
X

Figure 1: Two-dimensional solid body rotation and advection of a symmetric gaussian profile after (a,b,c) 50and (d,e,f) 300 rotations, calculated with (a,d) the MMA(34×34 grid points), (b,e) the CIP-CSL2 (42×42), and (c,f)
the backsubstitution (84×84) schemes.

show serious numerical diffusion after several tens ofrotation periods, the MMA scheme completely preserves the profile even after hundreds of rotation periods. This is a very important property for Vlasov
simulations of magnetized plasma. Note that since thenumbers of dependent variables are different among
the three schemes, we use the different number of grid
points so that the total memory usage is equal.
Applying the MMA scheme to the electromagnetic
Vlasov simulation, we simulate the one-dimensional,
strictly perpendicular collisionless shock waves. The
simulation parameters are as follows; a proton to electron mass ratio mp /me = 25, a ratio of the electron
plasma to gyro frequency ωpe /ωge = 100, electron
and proton plasma beta values βe = βp = 1.0, andthe spatial grid size ∆x = 20λD , where λD is the Debye length. Figure 2 shows the plasma phase space
distribution and the electromagnetic fields. An AlfvénMach number of the resulting shock wave is ∼ 7.5
measured in the shock rest frame. The simulation
describes fundamental properties of the perpendicularcollisionless shock such as the electrostatic shock potential and the so-called reflected ions, and satisfies theRankine-Hugoniot conservation laws.

4 SummaryBz/B00.4

Ex/Ey0-0.4

50.0 rotations

Ey/Ey0Y0.21.0e+02
1.0e+01
1.0e+00
1.0e+02
1.0e+01
1.0e+00
5.0e+02
5.0e+01
5.0e+00
5.0e+02
5.0e+01
5.0e+00864204
20-2-42
10-1-2-30

50

100
X/c/ωpe

150

200

Figure 2: One-dimensional electromagnetic Vlasov simulation of perpendicular shock waves. (a,b) The electron
phase space distributions in (x, vx ) and (x, vy ). (c,d) The
proton phase space distributions in (x, vx ) and (x, vy ).
(e,f,g) The electromagnetic field (Bz , Ex , Ey ).

sented. Although the grid size is much larger than theDebye length, the simulation is stable and the fundamental shock properties are well described. This advantage over an explicit PIC simulation enables us toperform large-scale plasma kinetic simulations.

Acknowledgement
Figure 1 is reproduced by permission of the Elsevier
Inc.

References[1] C. Z. Cheng and G. Knorr: Journal of Computational Physics 22 (1976) 330.
[2] T. Minoshima, Y. Matsumoto and T. Amano: Journal of Computational Physics 230 (2011) 6800.
[3] K. Takizawa, T. Yabe and T. Nakamura: Computer
Physics Communications 148 (2002) 137.

We have presented a new numerical scheme for solving the advection equation and the Vlasov equation. [4] H. Schmitz and R. Grauer: Computer Physics
The scheme has a quite high capability necessary for
Communications 175 (2006) 86.
the Vlasov simulation of magnetized plasma. The application of the scheme to the electromagnetic Vlasov
simulation of collisionless shock waves has been pre-