High order moment closure for Vlasov-Maxwell equations

Yana DI1,2,3, Zhenzhong KOU3, Ruo LI4
1 The State Key Laboratory of Scientific and Engineering Computing, Chinese Academy of
Sciences, Beijing 100190, China
2 National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of
Sciences, Beijing 100190, China
3 Academy of Mathematics and Systems Science, Chinese Academy of Sciences,
Beijing 100190, China
4 HEDPS & CAPT, LMAM & School of Mathematical Sciences, Peking University,
Beijing 100871, China
c
Higher Education Press and Springer-Verlag Berlin Heidelberg 2015
Abstract The extended magnetohydrodynamic models are derived based on
the moment closure of the Vlasov-Maxwell (VM) equations. We adopt the Grad
type moment expansion which was firstly proposed for the Boltzmann equation.
A new regularization method for the Grad’s moment system was recently
proposed to achieve the globally hyperbolicity so that the local well-posedness of
the moment system is attained. For the VM equations, the moment expansion
of the convection term is exactly the same as that in the Boltzmann equation,
thus the new developed regularization applies. The moment expansion of the
electromagnetic force term in the VM equations turns out to be a linear source
term, which can preserve the conservative properties of the distribution function
in the VM equations perfectly.
Keywords Moment closure, Vlasov-Maxwell (VM) equations, Boltzmann
equation, extended magnetohydrodynamics
MSC 65Z05
1 Introduction
Collisionless plasmas have been studied in a wide variety of fields, such as
in laboratory plasma physics, space physics, and astrophysics. Evolution of
collisionless plasmas and self-consistent electromagnetic fields are fully described
by the Vlasov-Maxwell (VM) equations. In this paper, we study the evolution
Received February 12, 2014; accepted February 12, 2015
Corresponding author: Yana DI, E-mail: yndi@lsec.cc.ac.cn
1088 Yana DI et al.
of a single species of nonrelativistic electrons under the self-consistent electromagnetic
fields while the ions are treated as uniform fixed background. Under
the scaling of the characteristic time by the inverse of the plasma frequency ω−1
p ,
length by the Debye length λD, and electric and magnetic fields by −mcωp/e
(with m the electron mass, c the speed of light, and e the electron charge), the
evolution of the distribution function f(t,x, v) of electrons is described by the
dimensionless form of the Vlasov equation
∂f
∂t
+ v · ∇xf + (E + v × B) · ∇vf = 0, x, v ∈ R3. (1)
The acceleration term in the phase space is the sum of the Coulomb force
and the Lorentz force, by which the particles interact via the electromagnetic
fields E and B. The charge and current densities, ρ and j, act as sources of
self-consistent electromagnetic fields according to the Maxwell equations:
∇x · E = ρ − ρi, ∇x · B = 0,
∇x × E = −∂B
∂t
, ∇x × B = j + ∂E
∂t
.
(2)
And they can be obtained by the moments of the particle distribution function,
ρ(t,x) =

f(t,x, v)dv, j(t,x) = ρ(t,x)u(t,x) =

vf(t,x, v)dv, (3)
respectively. The charge density of background ions is denoted by ρi, which is
chosen to satisfy total charge neutrality:

(ρ(t,x) − ρi)dx = 0.
The computation of the initial boundary value problem associated to the
VM equations is quite challenging, due to the high-dimensionality of the Vlasov
equation, multiple temporal and spatial scales associated with various physical
phenomena, nonlinearity, and the conservation of physical quantities. Particle-
In-Cell (PIC) methods (comprehensively reviewed in [19]) have been very
popular for a wide variety of plasma phenomena. Though the PIC method can
often give satisfying results, it inherently has the large statistical noise which
makes it difficult to study such as particle accelerations and thermal transport
processes, where a small number of high energy particles play an important
role. An alternative is represented by solving the Vlasov equation directly in
phase-space (namely, with a sixth dimensional computational grid in space and
velocity). It has been widely known that a numerical solution of the
advection equation suffers from spurious oscillations and numerical diffusion,
while a highly accurate scheme is required to preserve characteristics of the
Vlasov equation, saying the Liouville theorem, as much as possible. Though
there are a lot of studies [10,11,13,16,18] on the direct simulation, no standard
scheme for the Vlasov simulation has been established so far.
High order moment closure for Vlasov-Maxwell equations 1089
In this paper, we focus on the moment method where the distribution
function is expanded using Hermite functions in velocity space. The moment
method can be tracked back to Grad’s work in 1949 [6] for the Boltzmann
equation, where a 13-moment model was given as an extension of the classic
Euler equations. In [7], its major deficiencies were found soon, including the
appearance of subshocks in the structure of a strong shock wave and the loss
of global hyperbolicity. In the later study, a number of regularizations were
attempted to solve or alleviate these problems, such as Levermore’s work [9].
Jin and Slemrod [8] gave a regularization of the Burnett equations via
relaxation, which resulted in a set of equations containing the same variables
with Grad’s 13-moment theory, and no subshocks appeared in the structure of
shock waves. By integrating the moment method with the Chapman-Enskog
expansion, Struchtrup and Torrilhon [14,15] regularized Grad’s system to give
the R13 equations. The R13 system removes the discontinuities in the shock
wave and extends the region of hyperbolicity considerably [17].
The moment expansion for the Boltzmann equation can be extended to the
VM equations. The resulting convective term in the moment system expanded
from the drift term of the Vlasov equation has exactly the same format as that
of the Boltzmann equation. Thus, the method of the hyperbolic regularization
in [2] can be applied to the VM equations to achieve the global hyperbolicity.
The major difference of the VM equations from the Boltzmann equation is
the acceleration to the particles due to the electromagnetic fields. However, the
moment expansion of the acceleration term turns out to be a linear source term,
which can be expressed as a compact sparse coefficient matrix. Furthermore,
such a coefficient matrix is block diagonal for the moments with the same
order, which means that the evolutions of the moments with different orders
are separated. It is shown that a weighted l2 norm of the moments of the same
order is invariant in time accelerated by the magnetic field alone. As a result,
the high order moments are not growing at all due to the source term. So the
derived moment system is formulated as a quasi-linear system, plus a linear
source term which induces no growth of the high order moments. Since the
convection term in the system is guaranteed to be globally hyperbolic by the
regularization, the local well-posedness of the system is partially achieved.
The rest of this paper is arranged as follows. In Section 2, we present the
moment expansion of VM equations and regularize it to achieve the final hyperbolic
moment system. In Section 3, we discuss the conservation properties of
the moment system. And an exact VM equilibrium with sheath-like magnetic
field will be given in Section 4 for better understanding of the structure of the
derived moment system. Concluding remarks are in the last section.
2 Gradmomentsystem
When the density of electrons is not extremely high, we can assume that the
1090 Yana DI et al.
equilibrium distribution is a Maxwellian distribution:
feq(t,x, v) = ρ(t,x)
(2πkBT(t,x))3/2
exp

− (v − u(t, x))2
2kBT(t,x)

, (4)
where kB is the Boltzmann constant, and T(t,x) is the particle temperature,
which is related with the distribution function as follows:
3ρ(t,x)kBT(t,x) =

|v − u(t,x)|2f(t,x, v)dv. (5)
In this section, we derive the moment system of the Vlasov equation using
the Grad type moment expansion. The basic idea is to expand the distribution
function into an infinite series using Hermite functions as basis functions, and
write down the equation for each coefficient. By cut-off and closure, we obtain
a system with finite equations to approximate the Vlasov equation.
2.1 Hermite expansion of distribution function
Following the method in [3,4], we expand the distribution function into Hermite
series as
f(t,x, v) =

α∈N3
fα(t,x)HT ,α

v − u(t,x)
T (t,x)

, (6)
where fα(t,x) are the coefficients. The basis function HT ,α is an exponentially
decaying function multiplied by a multi-dimensional Hermite polynomial shifted
by the local macroscopic momentum u(t,x) and scaled by the square root of
the local temperature T (t,x) = kBT(t,x):
HT ,α(ξ) =
3
d=1
1 √
2π
T (t,x)−(αd+1)/2Heαd (ξd) exp

− ξ2
d
2

, (7)
where α is considered as a multi-index. The Hermite polynomials Hen(x) can
be obtained by successive differentiation with respect to the Gaussian function:
Hen(x) = (−1)n exp

x2
2
dn
dxn exp

− x2
2

. (8)
For convenience, Hen(x) is taken as zero if n < 0, and thus, HT ,α(ξ) is zero
when any component of α is negative. Naturally, the equilibrium distribution
feq(t,x, v) in (4) is coincidently equal to the first term of expansion. Using
the orthogonality of the Hermite polynomials, some simple relations can be
obtained:
ρ = f0, fei
≡ 0, i= 1, 2, 3,
3
d=1
f2ed = 0, (9)
qi = 2f3ei +
3
d=1
f2ed+ei, pij = δijρT + (1 + δij)fei+ej , (10)
High order moment closure for Vlasov-Maxwell equations 1091
where ed is the unit vector with its d-th entry to be 1, qi is the heat flux, and
pij is the pressure tensor.
2.2 Moment expansion of Vlasov equation
The general method to get the moment system is to multiply the Vlasov
equation (1) by polynomials of momentum v and integrate both sides over
v on R3. One equivalent way is as follows. We substitute the expansion of
the distribution function (6) into the Vlasov equation (1), then collect the
coefficients of the same order on both sides, and finally equate them to yield
the derived moment system. It should be noted that the Hermite function (7)
used in this paper depends also on the time t and position x through u(t,x)
and T (t,x), which is different from the general expansion using the Hermite
functions depending only on the momentum v [12]. For convenience, we list
some useful relations of Hermite polynomials as follows [1]:
(i) orthogonality:

R
Hel(x)Hen(x) exp

− x2
2

dx = l!
√
2π δl,n;
(ii) recursion relation:
Hen+1(x) = xHen(x) − nHen−1(x);
(iii) differential relation:
He

n(x) = nHen−1(x).
And the following equality can be derived from the last two relations:
∂
∂vj
HT ,α

v − √ u
T

= −HT ,α+ej

v√− u
T

. (11)
With these relations, the general moment equations can be obtained with a
slight rearrangement by matching the coefficients of the same weight function:
∂fα
∂t
+
3
d=1

∂ud
∂t
+
3
j=1
uj
∂ud
∂xj
− Ed −
3
k,m=1
εdkmukBm

fα−ed
−
3
d,k,m=1
εdkm(αk + 1)Bmfα−ed+ek
+
1
2


∂T
∂t
+
3
j=1
uj
∂T
∂xj
3
d=1
fα−2ed
+
3
j,d=1

∂ud
∂xj
(T fα−ed−ej + (αj + 1)fα−ed+ej )
1092 Yana DI et al.
+
1
2
∂T
∂xj
(T fα−2ed−ej + (αj + 1)fα−2ed+ej )


+
3
j=1

T
∂fα−ej
∂xj
+ uj
∂fα
∂xj
+ (αj + 1)
∂fα+ej
∂xj

= 0, (12)
where the Levi-Civita symbols εdkm are defined as
εdkm =
⎧⎪⎨
⎪⎩
1, d = k = m cyclic permutation of 1,2,3,
−1, d = k = m anti-cyclic permutation of 1,2,3,
0, (d − k)(k − m)(m − d) = 0.
By setting α = 0 in (12), we deduce the mass conservation:
∂ρ
∂t
+
3
j=1

uj
∂ρ
∂xj
+ ρ
∂uj
∂xj

= 0. (13)
By setting α = ed, with d = 1, 2, 3 and noting that fed = 0 in (12), we obtain
∂ud
∂t
+
3
j=1
uj
∂ud
∂xj
+
1
ρ
3
j=1
∂pjd
∂xj
= Ed +
3
k,m=1
εdkmukBm. (14)
By setting α = 2ed, we obtain
∂f2ed
∂t
+ ρ
2


∂T
∂t
+
3
j=1
uj
∂T
∂xj

+ ρT
∂ud
∂xd
+
3
j,l=1
(1 + 2δjd)f2ed−el+ej
∂ul
∂xj
−
3
k,m=1
εdkmBmfed+ek
+
3
j=1
uj
∂f2ed
∂xj
+ (1+2δjd)
∂f2ed+ej
∂xj
= 0. (15)
Noting that
3
d=1
f2ed = 0,
we sum the upper equations over d to get
ρ


∂T
∂t
+
3
j=1
uj
∂T
∂xj

+
2
3
3
j=1


∂qj
∂xj
+
3
d=1
pjd
∂ud
∂xj

= 0. (16)
Since
ρT =
1
3
3
d=1
pdd,
High order moment closure for Vlasov-Maxwell equations 1093
we have
∂T
∂xj
=
1
3ρ
3
d=1
∂pdd
∂xj
− T
ρ
∂ρ
∂xj
, j= 1, 2, 3. (17)
Substituting (14), (16), and (17) into (12), we eliminate the time derivatives of
u and T and the spatial derivatives of T . Then the quasi-linear form of the
moment system reads
∂fα
∂t
+
3
j=1

T
∂fα−ej
∂xj
+ uj
∂fα
∂xj
+ (αj + 1)
∂fα+ej
∂xj

+
3
j=1
3
d=1


T fα−ed−ej + (αj + 1)fα−ed+ej
− pjd
3ρ
3
k=1
fα−2ek

∂ud
∂xj
−
3
j=1
3
d=1
fα−ed
ρ
∂pjd
∂xj
− 1
3ρ
3
k=1
fα−2ek
3
j=1
∂qj
∂xj
+
3
j=1

  3
k=1
(T fα−2ek−ej + (αj + 1)fα−2ek+ej )


− T
2ρ
∂ρ
∂xj
+
1
6ρ
3
d=1
∂pdd
∂xj

=
3
d,k,m=1
εdkm(αk + 1)Bmfα−ed+ek , ∀ |α| 2. (18)
Remark 1 Taking ρ, ud (d = 1, 2, 3), pij (1 i j 3), and fα (|α| 3)
as unknowns, equations (13), (14), and (18) are collected to obtain a moment
system with an infinite number of equations. By the relation between ud and
fed given in (9) and the definition of qi and pij in (10), one can see that the
obtained system is quasi-linear for the unknowns.
2.3 Moment closure with global hyperbolicity
For a positive integer M 3, expansion (6) is truncated as
fh(t,x, v) =

|α| M
fα(t,x)HT ,α

v − u(t,x)
T (t,x)

. (19)
For any
α ∈ SM = {α ∈ N3 | |α| M},
let
N (α) =
3
i=1


3
k=4−i αk + i − 1
i

+ 1 (20)
be the ordinal number of α in SM. Then the total number of the set SM will
be
N = N (Me3) =


M + 3
3

.
1094 Yana DI et al.
Let w = (w1, . . . ,wN)T ∈ RN, for each i, j ∈ {1, 2, 3} and i < j,
w1 = ρ, wN (ei) = ui, (21a)
wN (2ei) = pii
2, wN (ei+ej) = pij , (21b)
wN (α) = fα, 3 |α| M. (21c)
The moment system (13), (14), and (18) is collected in a quasi-linear format as
∂w
∂t
+
3
j=1
Mj(w) ∂w
∂xj
= Gw + g, (22)
where Mj , G are N × N matrices, and g is N vector, corresponding to the
terms with derivatives of w, the magnetic force term, and the electronic force
term, respectively. Precisely, from (14) and (18), we have
gN (ei) = Ei, GN (α),N (α−ed+ek) =
3
m=1
εdkm(αk +1)Bm, (23)
while all other entries of G and g vanish.
It has been pointed out in [2] that it is not appropriate to set
∂fα+ej
∂xj
= 0 (|α| = M)
as the closure since the system is lack of hyperbolicity if the distribution function
is far away from the equilibrium. For any α with |α| = M, we define
RM
j(α) = (αj + 1)
3
d=1
fα−ed+ej
∂ud
∂xj
+
1
2

  3
d=1
fα−2ed+ej

∂T
∂xj

, (24)
and
ˆM
j
∂w
∂xj
=Mj
∂w
∂xj
−

|α|=M
RM
j(α)IN (α) (25)
for any admissible w, where Ik is the k-th column of the N ×N identity matrix.
We regularize system (22) as
∂w
∂t
+
3
j=1
ˆM
j(w) ∂w
∂xj
= Gw + g. (26)
We refer the readers to [2] for more details. The global hyperbolicity has been
rigorously proven in [2]. For convenience, the most important result is given in
the following lemma.
High order moment closure for Vlasov-Maxwell equations 1095
Lemma 1 The regularized moment system (26) is hyperbolic for any w with
positive temperature. Precisely, for a given unit vector n = (n1, n2, n3), the
matrix
3
j=1
nj ˆMj(w) (27)
is diagonalizable with real eigenvalues as
u · n + Ck,m
√
T , 1 k m M + 1, (28)
where Ck,m is a root of m-order Hermite polynomial, and satisfies
C1,m < C2,m < · · · < Cm,m.
The structure of the N eigenvectors can be fully clarified.
Remark 2 Based on Lemma 1, the regularized moment system (26) is
locally well-posed due to the hyperbolicity. We would like to mention that the
regularization here actually does not add any new terms to system (22). On the
contrary, it has erased the terms in (18) with a factor αj + 1 in its coefficient
for the equations of fα with |α| = M only.
3 Conservations
It is well known that the Vlasov-Maxwell equations, (1) and (2), conserve total
number of particles, momentum, and energy, which are given, respectively, by
N =

fdvdx, (29)
P =

vfdv + E × B

dx, (30)
E =
1
2

v2fdv + (E2 +B2)

dx. (31)
Due to the truncation (19), all the above conservations can be expressed by the
moments of first three orders:
Nh =

fhdvdx =

ρdx, (32)
Ph =


vfhdv + E × B

dx =

(ρu + E × B)dx, (33)
Eh =
1
2

v2fhdv + (E2 + B2)

dx
=
1
2

[ρ(3T + u2) + (E2 + B2)]dx. (34)
1096 Yana DI et al.
Actually, using equations (13), (14), and (16), it is straightforward to obtain
the following proposition.
Proposition 1 The semi-discrete moment expansion fh(t,x, v) has the
following conservation properties:
dNh
dt
= 0,
dPh
dt
= 0,
dEh
dt
= 0. (35)
Denoting by s → (t(s),x(s)) a parametric representation of the
characteristic integral curves associating with eigenvalue λ and left eigenvector
l, the characteristic equations become
lT dw
ds
= lT(Gw + g). (36)
So omitting the convective term

3
j=1Mj(w)dw/dxj in (22) temporarily, let
us consider the system with the source term only
dw
ds
= Gw + g. (37)
We first point out that the matrix G is block diagonal as
G = diag{0,G1,G2, . . . ,GM},
where
Gm = [GN (α),N (β)], |α| = |β| = m, 1 m M,
and the nonzero entries are given by (23). We define a partition diagonal matrix
D = diag{1,D1,D2, . . . ,DM},
where
Dm = diag{DN (α),N (α)
}|α|=m, DN (α),N (α) = α ! :=
3
d=1
αd!. (38)
Correspondingly, the vector w is divided into
w = [ρ,uT, ˆw
T2
, . . . , ˆwT
M]T,
where
ˆw
m = [wN (α)]T, |α| = m.
We have the following properties.
Proposition 2 The solution of system (37) satisfies
dρ
ds
= 0, (39)
d
ds
1
2 uTu

= uTE, (40)
d
ds
1
2
ˆw
T
mDmˆwm

= 0, 2 m M. (41)
High order moment closure for Vlasov-Maxwell equations 1097
Proof There is no source term for the density ρ, so (39) is obvious. In (23), it
is noticed that
3
i,k,m=1
εikmuiukBm = 0,
and thus, (40) is obtained. As for (41), we have
d
ds
1
2
ˆw
T
mDmˆwm

= ˆw T
mDm
∂ˆwm
∂s
= ˆwT
mDmGmˆwm.
Notice that the matrix DmGm is a diagonal block of the matrix DG, and from
(23), each one of its nonzero entries satisfies
(DG)N (α),N (α−ed+ek) = α !GN (α),N (α−ed+ek)
=

m=1,2,3
εdkm(αk + 1)Bmαd! αk! αm!
=

m=1,2,3
εdkmαdBm(αd − 1)! (αk + 1)! αm!
= −(α − ed + ek)!GN (α−ed+ek),N (α)
= −(DG)N (α−ed+ek),N (α).
It is turned out that DmGm is a skew-symmetric matrix, and thus,
ˆw
T
mDmGmˆwm = 0. (42)
This ends the proof.
Remark 3 The Coulomb force, which is the term g, provides an acceleration
on the mean velocity only, and the Lorentz force in G exerting on the mean
velocity is perpendicular to the mean velocity clearly. The result in Proposition
2 indicates that the Lorentz force will not change the magnitude of the high
order moments for any order m 2, too, taking the matrix Dm as the l2 weight.
One may observe that the Lorentz force in the Vlasov equation will rotate the
distribution function in the velocity space only, and here we see such behavior
is preserved in the moment system we derived.
4 Exact Vlasov-Maxwell equilibrium
Knowledge of the exact Vlasov-Maxwell equilibrium is often necessary when
analyzing the stability of a plasma [5]. In this section, we present a simple
example of exact Vlasov-Maxwell equilibrium. This makes it possible to
examine the residual of the moment system if we substitute the exact
solution into the moment system. The moment system we derived is then
partially validated once it is observed that the residual of the system is small
enough.
1098 Yana DI et al.
For simplicity, we consider a situation in which all quantities vary only in
the x1 direction and the magnetic field is unidirectional with one component
B3 in the x3 direction. Then the magnetic field can be derived from a potential,
A2,
B3 =
dA2
dx1
, B3(−∞) = B0. (43)
The equilibrium is characterized by a zero electric field. The Vlasov equation
(1) becomes
v1
∂f
∂x1
+ v2
dA2
dx1
∂f
∂v1
− v1
dA2
dx1
∂f
∂v2
= 0. (44)
The exact distribution function can be obtained (more details in [5]),
f(x1, v) = β2B2
0
8π2

πβ
2γδ

1/2
exp

− (v2 + A2(x1))2
4δ
− β|v|2
2

, (45)
where β and γ are constants, and
δ =
1
4γ
− 1
2β
.
Maxwell’s equations (2) for the magnetic field become
d2A2
dx21
= −B2
0γA2 exp(−γA22
). (46)
Let us calculate the moments and plug them into our moment equations.
Then we find that the residual is going to zero as the order goes to infinity.
With the expression (45), the density ρ(x1), the mean velocity u(x1), and the
temperature T (x1) are calculated:
ρ(x1) =

R3
f(t, x1, v)dv = β
2 B2
0 exp(−γA22
(x1)), (47)
u(x1) =
1
ρ

R3
vf(t, x1, v)dv =

0,
−2γ
β
A2(x1), 0

T
, (48)
T (x1) =
1
3ρ

R3
(v − u)2f(t, x1, v)dv =
1
β
− 2γ
3β2

. (49)
Then the equilibrium distribution function f(x, v) can be expanded into the
Hermite series:
f(x, v) =

α
fα(x)HT ,α

v − u(x)
T (x)

, (50)
where
fα(x) =
⎧⎪⎨
⎪⎩

2γ
3β2

α1/2
(α1)!!
−4γ
3β2

α2/2
(α2)!!

2γ
3β2

α3/2
(α3)!!
β
2 B2
0 exp(−γA22
(x1)), αi even,
0, otherwise.
(51)
High order moment closure for Vlasov-Maxwell equations 1099
It is obvious that
lim
|α|→+∞
fα(x) = 0. (52)
Actually, all the moment equations, which are not modified by the closure,
are satisfied by the moments of f(x, v). In the regularized moment system with
the truncation order M, it is only required to examine the moment equations
of order |α| = M, which has been modified due to the truncation and
closure. Substituting the exact moments into the regularized moment system
and calculating the residual yields
Res(α) =
3
j=1
Rj
M(α) +
3
j=1
(αj + 1)
∂fα+ej
∂xj
, (53)
where the closure term Rj
M(α) is defined in (24) and the truncation term
3
j=1
(αj + 1)
∂fα+ej
∂xj
is easy to be identified by observing the original moment equation (18). The
residue (53) is reduced into
Res(α) = (α1 + 1)
du2
dx1
fα−e2+e1 + ∂fα+e1
∂x1

=
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

2γ
3β2

(α1+1)/2
−4γ
3β2

(α2−1)/2

2γ
3β2

α3/2
(α1 − 1)!! (α2 − 1)!! α3!! ρ
du2
dx1
,
α = (2k1 − 1, 2k2 + 1, 2k3), ki > 0,
(T1 − T )(α1+1)/2(T2 − T )α2/2(T3 − T )α3/2
(α1 − 1)!! α2!! α3!!
dρ
dx1
,
α = (2k1 − 1, 2k2, 2k3), ki > 0,
0, otherwise.
The residue goes to zero as the truncation order M going to infinity, i.e.,
lim
|α|→+∞
Res(α) = 0. (54)
5 Conclusion
In summary, the hyperbolic moment system has been derived for the Vlasov-
Maxwell equations. We have proven that the moment method conserves charge,
conserves momentum, conserves energy and is stable. The corresponding
numerical method is developing for the moment system obtained, and will be
applied to study important plasma physics problems in the future.
1100 Yana DI et al.
Acknowledgements The research of Y. Di was supported in part by the National
Magnetic Confinement Fusion Science Program (2011GB105003) and the National Natural
Science Foundation of China (Grant No. 11271358). The research of R. Li was supported
in part by the Sci-Tech Interdisciplinary Innovation and Cooperation Team Program of the
Chinese Academy of Sciences and the National Natural Science Foundation of China (Grant
Nos. 11325102, 91330205).
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