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SYMMETRIC AND SYMPLECTIC
METHODS FOR GYROCENTER
DYNAMICS IN TIME-INDEPENDENT
MAGNETIC FIELDS
by
Beibei Zhu, Zhenxuan Hu, Yifa Tang and Ruili Zhang
Report No. ICMSEC-15-3
November 2015
Research Report
Institute of Computational Mathematics
and Scientific/Engineering Computing
Chinese Academy of Sciences
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International Journal of Modeling, Simulation, and Scientific Computing
c World Scientific Publishing Company
⃝
SYMMETRIC AND SYMPLECTIC METHODS FOR
GYROCENTER DYNAMICS IN TIME-INDEPENDENT
MAGNETIC FIELDS
Beibei Zhu
LSEC, ICMSEC, Academy of Mathematics and Systems Science
Chinese Academy of Sciences, Beijing 100190, P.R. China
[email protected]
Zhenxuan Hu
School of Software, Fudan University, Shanghai 201203, P.R. China
[email protected]
Yifa Tang∗
LSEC, ICMSEC, Academy of Mathematics and Systems Science
Chinese Academy of Sciences, Beijing 100190, P.R. China
[email protected]
Ruili Zhang
Department of Modern Physics and School of Nuclear Science and Technology
University of Science and Technology of China, Hefei, 230026, P.R. China
[email protected]
Received (Day Month Year)
Accepted (Day Month Year)
We apply a second order symmetric Runge-Kutta method and a second order symplectic Runge-Kutta method directly to the gyrocenter dynamics which can be expressed
as a non-canonical Hamiltonian system. The numerical results show the overwhelming superiorities of the two methods over a higher-order non-symmetric non-symplectic
Runge-Kutta method in long-term tracking ability and near energy conservation. Furthermore, they are much faster than the midpoint rule applied to the canonicalized
system to reach given precision.
Keywords: Symmetric Runge-Kutta Method; Symplectic Runge-Kutta Method; Numerical Accuracy; Near Energy Conservation
1. Introduction
The gyrocenter dynamics of charged particles in time-independent magnetic fields is
a non-canonical Hamiltonian system. We apply a second order symmetric RungeKutta method and a second order symplectic Runge-Kutta method directly to
∗ Corresponding
author
1
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B. Zhu et al.
the dipole magnetic field and the Tokamak magnetic field, the numerical results
show the overwhelming superiorities of the two methods over a higher-order nonsymmetric non-symplectic Runge-Kutta method in long-term tracking ability and
and near energy conservation. Furthermore, we show that the first two methods
are much faster than the midpoint rule applied to the canonicalized Hamiltonian
system.
The dynamics of charged particle in magnetized plasma has two components:
the fast gyromotion and the slow gyrocenter motion, which rises as a multi-scale
problem. One way to resolve the problem is separating the fast gyromotion from the
slow guiding center motion. The gyrokinetic theory has been developed1,2,3 based on
this idea. More generally, gyrokinetics is a useful tool for simulating low-frequency
microinstabilities in magnetized plasma. After elimination of the high-frequency
elements in the formulation, the gyrokinetic equations usually form a non-canonical
Hamiltonian system. Then, the numerical integrators can be adopted for simulating
the system.
Standard numerical integrators, such as the high-order Runge-Kutta methods, guarantee small error for each single time step. However, they are incapable
to prevent error from accumulating over long time integration, which results in
large deviations even for sufficiently small step-sizes. Symmetric/revertible methods have been investigated by many researchers and shown to have some good
properties which are desired for numerical experiment 4,5 . Symplectic integrators,
well-known for their long-term tracking ability and global conservation properties,
can maintain the numerical accuracy and nearly preserve the energy of the system
in long time numerical simulation 5,6,7,8,9,10,11,12 . For non-canonical Hamiltonian
systems, since the standard symplectic integrators are not applicable, a common
way is to transform them into canonical ones by using coordinate transformations
13,14,15,16
, then apply integrators for the canonicalized systems to perform numerical experiments, or equivalently, directly construct symplectic numerical methods
via generating functions technique 17,18 . Such canonicalization still has two major
drawbacks: difficulty in obtaining the coordinate transformation and high computational complexity in numerical simulation. So it is preferable to use symmetric
methods and sympletic methods directly to simulate the gyrocenter dynamics without canonicalization.
In the literature, we will apply a second order symmetric Runge-Kutta method
and a second order symplectic Runge-Kutta method directly to the non-canonical
system to simulate the particle’s motion. We will show the superior long-term tracking ability and near energy-preserving property of the first two methods, in comparison with a standard third order Rung-Kutta method. Furthermore, the CPU
times of the two methods are found to be much less than the midpoint rule applied
to the canonicalized system.
The paper is organized as follows. A brief introduction to the gyrocenter system and its non-canonical-coordinate expressions are given in Section 2. In Section 3, the conceptions and properties of symmetric Runge-Kutta methods and
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symmetric and symplectic methods for gyrocenter dynamics in time-independent magnetic fields
3
symplectic Runge-Kutta methods are displayed. Section 4 presents all the numerical results for comparisons, generated by applying a second-order symmetric nonsymplectic Runge-Kutta method, a second-order symplectic non-symmetric RungeKutta method, a third-order non-symmetric non-symplectic Runge-Kutta method
to the non-canonical system, and by applying the midpoint rule to the canonicalized
system. Finally in Section 5, we give a brief summary of this work.
2. Gyrocenter dynamics
The Lagrangian of the gyrocenter system has the expression2
1
L(X, Ẋ, u, u̇) = [A(X) + ub(X)] · Ẋ − [ u2 + µB(X) + ϕ(X)],
2
where X represent the guiding center position, µ is the magnetic moment, u is the
parallel velocity of the guiding center and ϕ(X) is the scalar potential. A(X) =
⊤
(A1 , A2 , A3 ) is the vector potential which is normalized by cm/e, B(X) = ∇ × A
is the magnetic field, B(X) = |B| and b(X) = B/B = (b1 , b2 , b3 )⊤ is the unit
vector along the direction of the magnetic field. The gyrocenter equation is given
by the Euler-Lagrange equations of L with respect to X = (x, y, z)⊤ and u
K(v)v̇ = ∇H(v),
(1)
where v = (X⊤ , u)⊤ , H is a Hamiltonian, given by H(v) = 12 u2 + µB(X) + ϕ(X)
and K(v) is an antisymmetric matrix

0
 −a12
K(v) = 
 −a13
b1
a12
0
−a23
b2
a13
a23
0
b3

−b1
−b2 

−b3 
0
with
∂A2
∂A1
∂b2
∂b1
−
) + u(
−
),
∂x
∂y
∂x
∂y
∂A3
∂A1
∂b3
∂b1
=(
−
) + u(
−
),
∂x
∂z
∂x
∂z
∂A3
∂A2
∂b3
∂b2
=(
−
) + u(
−
).
∂y
∂z
∂y
∂z
a12 = (
a13
a23
It is easy to check that
2
det(K(v)) = |a13 b1 − a13 b2 + a12 b3 | .
If det(K(v)) ̸= 0, Equation. (1) can be rewritten as a non-canonical Hamiltonian
system
v̇ = K −1 (v)∇H(v)
(2)
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B. Zhu et al.
with

0

1
 b3
K −1 (v) = √
det(K)  −b2
−a23
−b3
0
b1
a13
b2
−b1
0
−a12

a23
−a13 
.
a12 
0
Here K(v) = (klm ) is a non-degenerate matrix which satisfies the Jacobi identity
∂klm
∂kmn
∂knl
+
+
= 0,
∂vn
∂vl
∂vm
1 ≤ l, m, n ≤ 4.
3. Symmetric and symplectic methods
We consider the system of differential equation
Ż = f (Z),
Z ∈ R2n .
(3)
Symmetry is an important property of numerical methods which is highly relevant to the order of accuracy and the geometric properties of the solution.
Definition 1
revertible) if
(see Refs. 4-5) A numerical method Φτ is called symmetric (or
Φ−τ ◦ Φτ = id.
Denoting by φτ (Z0 ) the phase flow of equation (3), i.e. the exact solution Z(τ )
after one time step τ with initial condition Z(0) = Z0 . The numerical flows Φτ
obtained by numerical methods for (3) can approximate φτ for sufficiently small
step-size τ . A method has order p if it satisfies the following formula:
Φτ (Z) = φτ (Z) + O(τ p+1 ).
It is well known that symmetric methods have even order.
Symplectic methods have a global conservative property for Hamiltonian systems which guarantees that the energy error is bounded by a small number for
sufficiently small time steps. For canonical Hamiltonian system
Ż = J
−1
(
∇H(Z),
J=
0
−In
)
In
,
0
a numerical method Φτ : Z → Z̃ is symplectic if and only if it satisfies
[
]⊤ [
]
∂ Z̃
∂ Z̃
J
= J,
∂Z
∂Z
(4)
(5)
for any sufficient small step-size τ and any Hamiltonian H. It is well known that
symplectic methods can preserve the structure of the Hamiltonian system (4).
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symmetric and symplectic methods for gyrocenter dynamics in time-independent magnetic fields
5
Therefore it is a common way to apply symplectic methods to simulate Hamiltonian systems. But in this literature, we choose to use symplectic methods to
non-canonical Hamiltonian system.
For non-canonical Hamiltonian system (2), the phase flow φτ (v) is a oneparameter group of K-symplectic transformations with the definition as follows:
[ τ
]T
[ τ
]
∂φ (v)
∂φ (v)
τ
K(φ (v))
= K(v).
∂v
∂v
K-symplectic methods are usually very difficult to find for (2). According to Darboux’s theorem 19,20 , there exists a coordinate transformation Z = ψ(v) that
transforms (2) into a canonical Hamiltonian system like (4) with new Hamiltonian
H̃ = H ◦ ψ −1 . Many researcher were devoted to canonicalizing the non-canonical
Hamiltonian system (2) (see Refs. 14-16). But it is not an easy task to acquire the
corresponding coordinate transformation. And symmetric and symplectic methods
had been shown to have good behavior in simulating the Ablowitz-Ladik model
which is a non-canonical Hamiltonian system (see Ref. 21). So we want to apply
symmetric methods and symplectic methods directly to (2) for numerical simulation.
3.1. Symmetric and symplectic Runge-Kutta methods
Runge-Kutta schemes22,23 form an important class of numerical methods for integrating the ordinary differential equations. An s-stage Runge-Kutta method is
given by
Ze = Z + τ
Ki = Z + τ
s
∑
i=1
s
∑
bi f (Ki ),
(6)
aij f (Kj ),
j=1
where bi and aij are real numbers and τ is the step-size. The properties of a RungeKutta method, such as the symmetry, the symplecticity and its order, are determined by its coefficients.
An algebraic characterization of symmetric Runge-Kutta methods was first presented by Stetter 4 .
Theorem 1 (see Ref. 4) A Runge-Kutta method is symmetric if and only if there
exists a permutation σ of {1, 2, . . . , s} such that for 1 ≤ i, j ≤ s,
bσ(i) = bi ,
bσ(j) − aσ(i)σ(j) = aij .
The systematic study of symplectic Runge-Kutta methods started around 1988,
and its algebraic characterization has been found7,8,9 . Symplectic Runge-Kutta
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B. Zhu et al.
methods are widely used among symplectic methods because of their simple form
and their stability.
As is well know that for Hamiltonian systems, a Runge-Kutta method (6) is
symplectic if
bi bj − bi aij − bj aji = 0, 1 ≤ i, j ≤ m.
4. Numerical simulations
In this section, a second order symmetric Runge-Kutta method, a second order
(canonically) symplectic Runge-Kutta method and a third order non-symmetric
and non-symplectic Runge-Kutta method are applied to simulate the dynamics of
gyrocenters in the dipole magnetic field and in the Tokamak magnetic field. In
addition, the CPU time of the midpoint rule applied to the canonicalized system is
compared with that of the first two methods.
Scheme 1: a second order symmetric non-symplectic Runge-Kutta method 21 .
In this paper, this method is used to demonstrate that all the benefits in the simulations only come from symmetry. The method is displayed as follows:

1
1


vn+1 = vn + τ f (W1 ) + τ f (W2 ),


2
2


1
1
(7)
W1 = vn + τ f (W1 ) + τ f (W2 ),

6
4




 W2 = vn + 1 τ f (W1 ) + 1 τ f (W2 ).
4
3
Scheme 2: a second-order symplectic non-symmetric Runge-Kutta method 21 .
This method is used to exclude the influence of symmetry and demonstrate the
superiority of the symplecticity.

3
1


vn+1 = vn + τ hf (W1 ) + τ f (W2 ),


4
4


1
3
(8)
W1 = vn + τ f (W1 ) + τ f (W2 ),

8
16




 W2 = vn + 3 τ f (W1 ) + 3 τ f (W2 ).
16
8
Scheme 3: a third-order non-symmetric and non-symplectic Runge-Kutta
method 16 . This method is of higher order than the above two schemes, so it is
fair to make comparisons. The expression of this method is displayed here:

1
1


vn+1 = vn + τ f (W1 ) + τ f (W2 ),


2
2√



3
3
(9)
W1 = vn + τ f (W1 ) −
τ f (W2 ),

6
6

√




 W2 = vn + 3 τ f (W1 ) + 3 τ f (W2 ).
6
6
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symmetric and symplectic methods for gyrocenter dynamics in time-independent magnetic fields
Method
CPU time (s)
symmetric Runge-Kutta
32.85
symplectic Runge-Kutta
31.32
7
Midpoint Rule
80.48
Scheme 4: the midpoint rule 16 . It should be noticed that this method is applied
to the canonicalized system. This method is of one stage which is simpler than
Schemes 1 and 2. To compare the CPU time of this method with that of Schemes
1 and 2, if the latter ones are less than the former one, the superiority of Schemes
1 and 2 will be more obvious.
(
)
Zn+1 + Zn
Zn+1 = Zn + τ f
.
(10)
2
4.1. Dipole magnetic field
In this subsection, we present the non-canonical-coordinate expression of the dipole
magnetic field. We choose the vector potential to be
(
)
My Mx
A(X) =
,− 3 ,0 ,
(11)
r3
r
√
where r = x2 + y 2 + z 2 and M is a constant. Via simple calculation, we have the
magnetic field B(X), the field strength B(X) and the unit magnetic field b(X):
(
)
3xz
3yz
2z 2 − x2 − y 2
B(X) = −M 5 , −M 5 , −M
,
r
r
r5
√
r r2 + 3z 2
(12)
B(X) = M
,
5
r
)
(
3xz
3yz
2z 2 − x2 − y 2
b(X) = − √
,− √
,− √
.
r r2 + 3z 2
r r2 + 3z 2
r r2 + 3z 2
The comparison of the particle’s motion in dipole magnetic field calculated by
Scheme 1 and 2 with Scheme 3 is shown in Fig. 1. We set the constant M to be
1000, and µ be 0.01. The initial condition v0 is chosen to be [1, 1, 1, 0.01]. It can
be seen that the obtained orbits are accurate over long time for Scheme 1 and
2, but extending outwards for Scheme 3. The numerically resulted values of the
Hamiltonian H at discrete temporal points are nearly identical for Scheme 1 and
2, but decrease without bound for Scheme 3.
We have also displayed in Fig. 2 the orbit and the scaled energy ratio (H/H0 )
of Scheme 4 applied to the canonical system with the coordinate transformation
given in 16 and the numerical result is satisfactory. With the same step-size τ and
same number of steps, we have calculated the CPU time of Scheme 1 and 2 applied
to the non-canonical system and Scheme 4 applied to the canonical system. One
observes from Table 1 that the CPU time of Scheme 4 is about three times of that
of the former two schemes.
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1.5
1.5
Scheme 2
1
1
0.5
0.5
Y
Y
Scheme 1
0
0
−0.5
−0.5
−1
−1
−1.5
1.4
1.6
1.8
2
2.2
2.4
2.6
−1.5
1.4
2.8
1.6
1.8
X
2
2.2
2.4
2.6
2.8
X
(a)
(b)
1.5
5
scheme 1
scheme 2
scheme 3
Scheme 3
0
0.5
−5
(H/H0−1)*10
Y
2
1
0
−10
−0.5
−15
−1
−20
−1.5
1.4
1.6
1.8
2
2.2
2.4
2.6
X
2.8
−25
0
500
1000
1500
2000
2500
3000
t/s
(c)
(d)
Fig. 1. The numerical results in dipole field. Fig. 1 (a) is the orbit obtained by the symmetric
Runge-Kutta method, Fig. 1 (b) by the symplectic Runge-Kutta method and Fig. 1 (c) by the
standard third-order Runge-Kutta method. Fig. 1 (d) display the evolution of the Hamiltonian H
of the three schemes where H0 represent the initial energy. The time-step size is τ = 0.05 and the
number of steps is N=60000.
4.2. Tokamak magnetic field
In this subsection, the above four methods are employed to solve the non-canonical
Hamiltonian system (2) in Tokamak magnetic field with two different initial conditions which lead to the two well-known particle’s orbits: the banana orbit and the
transit orbit.
The vector potential is chosen to be
A=
B0 r 2
R R0 B0
B0 R0 z
eζ − ln( )
ez +
eR ,
2Rq
R0
2
2R
(13)
√
where R = x2 + y 2 , and B0 , R0 and q are constants. One easily writes out the
magnetic field B(X), the field strength B(X) and the unit magnetic field b(X) as
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symmetric and symplectic methods for gyrocenter dynamics in time-independent magnetic fields
1.5
9
0.2
Scheme 5
0.1
1
0
0.5
(H/H0−1)*10
Y
4
−0.1
0
−0.5
−0.2
−0.3
−0.4
−0.5
−1
−0.6
−1.5
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
−0.7
0
5
X
10
15
20
25
30
35
40
t/s
(a)
(b)
Fig. 2. The orbit and the scaled energy ratio (H/H0 ) obtained by the midpoint rule applied to
the canonicalized system. The time-step size is τ = 0.01.
Method
CPU time (s) (Banana)
CPU time (s) (Transit)
symmetric Runge-Kutta
30.15
30.48
symplectic Runge-Kutta
29.83
31.97
Midpoint Rule
77.75
77.88
follows
B0 r
B0 R0
B0 r2
eθ +
eζ =
∇θ − B0 R0 ∇ζ,
qR
R
qR
√
B0
r2 + R02 q 2 ,
B(x) =
qR
−xz − R0 qy
−yz + R0 qy
R − R0
b(x) = ( √
, √
,√
).
2
2
2
2
2
2
R r + R0 q R r + R0 q
r2 + R02 q 2
B(x) =
(14)
We apply Schemes 1 and 2 to the original system to do numerical simulation.
Meanwhile, Scheme 3 is compared with the above two schemes. In these examples,
B0 and R0 are chosen to be 1 and q to be 2. Comparison of numerical results for
the banana orbit and energy evolution obtained by the above three methods are
displaced in Fig. 3. The initial condition for v is [1, 0, 0, 0.0004306]⊤ and that for
µ is 2.25 × 10−6 . The orbits by Scheme 1 and 2 in Fig. 3 (a) and (b) are both
closed and accurate over long time but in Fig. 3 (c) the orbit by Scheme 3 is
extending outwards and losing the accuracy. Fig. 4 displays the comparison of the
transit orbits calculated by the above three schemes with the initial conditions v =
[1.05, 0, 0, 0.0008117]⊤ , µ = 2.448×10−6 . The topological structures of the obtained
transit orbits are preserved perfectly by Schemes 1 and 2 but badly destroyed
by Scheme 3. Fig. 3 (d) and Fig. 4 (d) are the evolution of the energy of the
system obtained by the three schemes and they demonstrate that the corresponding
Hamiltonian is preserved very well by Schemes 1 and 2 but increased with time by
Scheme 3.
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0.08
0.08
Scheme 2
0.06
0.04
0.04
0.02
0.02
Y
Y
Scheme 1
0.06
0
0
−0.02
−0.02
−0.04
−0.04
−0.06
−0.06
−0.08
1
1.02
1.04
1.06
1.08
1.1
−0.08
1
1.02
1.04
X
1.06
1.08
(a)
(b)
7
0.08
scheme 1
scheme 2
scheme 3
Scheme 3
6
0.04
5
0.02
4
3
0.06
(H/H0−1)*10
Y
1.1
X
0
−0.02
3
2
1
−0.04
−0.06
0
−0.08
−1
1
1.02
1.04
1.06
X
(c)
1.08
1.1
0
0.5
1
1.5
2
2.5
t/s
3
3.5
4
4.5
5
6
x 10
(d)
Fig. 3. The numerical results of banana orbit in Tokamak magnetic field. Fig. 3 (a) is the orbit
obtained by the symmetric Runge-Kutta method, Fig. 3 (b) by the symplectic Runge-Kutta
method and Fig. 3 (c) by the standard third-order Runge-Kutta method. Fig. 3 (d) display the
scaled energy ratio (given on a exponential scale) of the three methods where H0 is the initial
energy. The time-step size is τ = 120 ∼ T /333.
We also use Scheme 4 to simulate the banana orbit and the transit orbit in the
canonicalized system and the numerical results in Fig. 5 are favorable. We have
calculated the CPU time of Schemes 1, 2 and 4 with the same step-size and same
number of steps, we find that for both orbits, the CPU times of Scheme 4 are much
longer than those of Scheme 1 and 2 (refer to Table 2).
5. Conclusion
In this paper, we have applied a symmetric Runge-Kutta method and a symplectic
Runge-Kutta method directly to the non-canonical system of gyrocenter dynamics.
In the cases of dipole magnetic field and Tokamak magnetic field as examples, we
demonstrate the overwhelming advantages of the two methods. On one hand, the
symmetric Runge-Kutta scheme and the symplectic Runge-Kutta scheme have the
significant superiority over the standard third-order Runge-Kutta scheme in longterm tracking ability and near energy conservation. On the other hand, the first
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symmetric and symplectic methods for gyrocenter dynamics in time-independent magnetic fields
0.06
0.06
Scheme 2
0.04
0.04
0.02
0.02
Y
Y
Scheme 1
0
0
−0.02
−0.02
−0.04
−0.04
−0.06
0.92
11
0.94
0.96
0.98
1
1.02
1.04
−0.06
0.92
1.06
0.94
0.96
0.98
X
1
1.02
1.04
1.06
X
(a)
(b)
0.06
3
Scheme 3
scheme 1
scheme 2
scheme 3
2.5
0.04
2
(H/H0−1)*10
Y
2
0.02
0
1.5
1
−0.02
0.5
−0.04
−0.06
0.92
0
0.94
0.96
0.98
1
X
(c)
1.02
1.04
1.06
−0.5
0
0.5
1
1.5
t/s
2
2.5
6
x 10
(d)
Fig. 4. The numerical results of the transit orbit in Tokamak magnetic field. Fig. 4 (a) is the
orbit obtained by the symmetric Runge-Kutta method, Fig. 4 (b) by symplectic Runge-Kutta
method and Fig. 4 (c) by the standard third-order Runge-Kutta method. Fig. 4 (d) display the
evolution of the Hamiltonian value H. The time-step size is τ = 120 ∼ T /250.
two methods are much faster than the midpoint rule applied to the canonicalized
system. Therefore, we conclude that the symmetric Runge-Kutta method and the
symplectic Runge-Kutta method are suitable for simulating the particles’ motions
of gyrocenter dynamics.
6. Acknowledgements
This research is supported by the ITER-China Program (Grant No. 2014GB124005)
and by the National Natural Science Foundation of China (Grant Nos. 11371357
and 11505186).
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