A numerical method for solving Hamiltonian equations is said to be symplectic if it preserves the symplectic structure associated with the equations. Various symplectic methods are widely used in many fields of science and technology. A symplectic method preserves an approximate Hamiltonian perturbed from the original Hamiltonian. It theoretically supports the effectiveness of symplectic methods for longterm integration. Although it is also related to longterm integration, numerical stability of symplectic methods have received little attention. In this paper, we consider explicit symplectic methods defined for Hamiltonian equations with Hamiltonians of the special form, and study their numerical stability using the harmonic oscillator as a test equation. We propose a new stability criterion and clarify the stability of some existing methods that are visually based on the criterion. We also derive a new method that is better than the existing methods with respect to a CourantFriedrichsLewy condition for hyperbolic equations; this new method is tested through a numerical experiment with a nonlinear wave equation.
I. INTRODUCTION
A symplectic integration method is a numerical method for solving Hamiltonian equations,
a special class of differential equations related to classical mechanics and symplectic geometry. Various symplectic methods are designed and widely used in celestial mechanics, molecular dynamics, electromagnetic field analysis, etc., particularly for the longterm integration of Hamiltonian equations.
The time evolution of Hamiltonian equations preserves a special differential 2form
dp
∧
dq
called the symplectic form. A numerical method is said to be symplectic if it also preserves the symplectic form. Since the concept of symplectic integration methods was proposed in the mid 1980s
[1]
, many mathematical researches have been carried out
[2

4]
. In particular, it has been revealed that a symplectic method preserves an approximate Hamiltonian perturbed from the original Hamiltonian
[5
,
6]
. It theoretically supports the effectiveness of symplectic methods for longterm integration.
On the other hand, the numerical stability of symplectic methods has received little attention, although it is also related to longterm integration; only a few papers
[7
,
8]
deal with this subject. It is certain that many outstanding symplectic methods are implicit and possess originally superior stability. However, in a largescale computation, e.g., in the solution of partial differential equations, explicit methods are still effective tools. A study of their stability has significance for practical computation because the stability of numerical methods is closely related to step size restrictions, such as a CourantFriedrichsLewy (CFL) condition for hyperbolic equations.
In this paper, we study the stability of an explicit symplectic method by using the harmonic oscillator as a test equation, following
[8]
. An outline of this paper is as follows: In Section II, we describe the fundamental concept and notation concerning explicit symplectic methods and their numerical stability. In Section III, we propose a new stability criterion for the symplectic methods and discuss the stability of the basic methods on the basis of this criterion. In Section IV, we continue to analyze more advanced methods and derive a new method, which is tested through a numerical experiment with the sineGordon equation, a nonlinear wave equation in Section V.
II. PRELIMINARIES
 A. Explicit Symplectic Methods
We consider a Hamiltonian of the special form
and the initial value problem
for the corresponding Hamiltonian equation, where
In mechanics,
T
and
U
represent kinetic energy and potential energy, respectively.
In general, symplectic methods are implicit; i.e., it is necessary to solve nonlinear equations for the implementation of these methods. For problem (3), there are explicit symplectic methods by virtue of the special form (2). A wellknown instance is a symplectic partitioned RungeKutta method, whose general form is as follows (see, e.g.,
[2
,
4]
):
Here, Δ
t
> 0 is the time step size,
t_{n}
=
nΔt
(
n
= 0,1,… ), and
q_{n}
p_{n}
are approximate values for
q(t_{n})
and
p(t_{n})
, respectively. Further,
b
_{1}
,
b
_{2}
, …
b
_{s}
,
and
are parameters of the method, and
Q_{i}
and
P_{i}
are intermediate variables for computation. The parameters of the method, determined from order conditions
[2
,
4]
, are often written as
 B. Test Equation for Stability Analysis
To study the stability of the symplectic method (5), we adopt the harmonic oscillator
as a test equation (
[8]
; see also
[7]
for another test equation). This is a Hamiltonian equation with the Hamiltonian
H(p,q)
= (ω /2 )(
p
^{2}
+
q
^{2}
), ω ≥ 0. We also adopt the scaled step size
as a basic parameter for the stability analysis. Upon the restriction of the frequency ω ≥ 0, the range of the parameter is
θ
≥ 0.
It should be noted that exact solutions to (6) satisfy
The matrix
M
(
≥
) is an orthogonal matrix, and its eigenvalues are
, both of which have unit modulus.
In the case
f
(
p
) =
ωp
and
g
(
t,q
) = −ω
q
, the equations for the intermediate variables in (5) becomes
The substitution of the first equation into the second equation gives
Hence, (9) is rewritten as
and application of method (5) to test equation (6) yields an analogue to (8),
It is clear that det
M_{i}
(
θ
) = 1. Hence
M
_{*}
(
θ
) = 1 holds for any method of the form(5). The Characteristic equation
M
_{*}
(
θ
) is written as
and the eigen values are given by
where tr
M
_{*}
(
θ
) denotes the trace of the matrix
M
_{*}
(
θ
). If │ tr
M
_{*}
(
θ
)│ < 2, the eigenvalues are complex numbers with λ=1. If tr
M
_{*}
(
θ
) = 2, then λ = 1, and if tr
M
_{*}
(
θ
) = − 2, then λ = − 1. If │tr
M
_{*}
(
θ
)│ > 2, the eigenvalues are real, and one of them satisfies λ > 1. The set {
θ
≥ 0 : │ tr
M
_{*}
(
θ
)│ ≤ 2} is a union of closed intervals. The connected component of the set that contains the origin is called the
stability interval
of method (5), which has been used for comparing the stability of numerical methods
[8]
.
III. STABILITY CRITERION
If │tr
M
_{*}
(
θ
)│ < 2,
M
_{*}
(
θ
) has complex conjugate eigenvalues λ,
which satisfy │λ│=│
│= 1 and λ ≠
Hence,
M
_{*}
(
θ
) is represented in form
with some nonsingular matrix
T.
Since
and │λ│=│
│ = 1, we have ║
M
_{*}
(
θ
)
^{n}
║ ≤║
T
║ ║
T
^{1}
║ for any integer
n
≥ 0, where ║•║ denotes the matrix norm induced from the Euclidean norm. The upper bound ║
T
║║
T
^{1}
║ is represented as follows.
Theorem 1.
Let
a,b,c,d
be real numbers, Assume that
satisfies det
M
= 1 and │ tr
M
│ < 2. Then, we have
for any integer
n
≥ 0, where
The proof of the theorem is obtained by a simple but tiresome computation. We omit the proof (cf. the proof of Theorem 3.1 in
[9]
). As shown below,
ϕ
in Theorem 1 is used as a criterion for the stability of the numerical methods.
In the case
s
= 1 and
(5) is reducde to
This is called the symplectic Euler method and is of the order 1 in accuracy. In the case of the symplectic Euler method, we have
Since tr
M
(
θ
) = 2 
θ
^{2}
, the stability interval of the method is [0, 2]. For 0 <
θ
< 2,
ϕ
in Theorem 1 is computed as
In the case
s
= 2, method (5) is rewritten as
which is of the order 2 if the parameters satisfy
In particular, the parameter values
satisfy the condition, and the corresponding method is known as the Störmer  Verlet method
[4
,
8]
.
For this method, we have
Since tr
M
(
θ
) = 2 −
θ
^{2}
, the stability interval of the Stormer  Verlet method is [0, 2], which is the same as that of the symplectic Euler method. However, since (2
θ

θ
^{3}
/4)
^{2 }
 {4  (2 
θ
^{2}
)
^{2}
} =
θ
^{6}
/16 , we have, for 0 < θ < 2,
Fig. 1
shows the functions
ϕ
for the two methods. Function (25) for the StörmerVerlet method is closer to the line
ϕ
= 1 than (20) for the symplectic Euler method. The matrix
M
(
θ
) in (8) is an orthogonal matrix and satisfies ║
M
(
θ
)
^{n}
║ = 1 for any
θ
≥ 0 and any integer
n
≥ 0. Since (25) reflects this property more appropriately than (20), we can consider the StörmerVerlet method has a better stability property than the symplectic Euler method although the two methods have the same stability intervals.
Table 1
presents
ϕ
and
ϕ
_{100}
= max
_{0≤n ≤100}
║
M
_{*}
(
θ
)
^{n}
║, computed numerically, for several values of
θ
. This shows that
ϕ
gives an appropriate approximation to sup
_{n≥0}
║
M
_{*}
(
θ
)
^{n}
║ except
θ
= 1
Functions ϕ for the symplectic Euler and StörmerVerlet methods.
Comparison betweenϕandϕ= max0≤n≤100║M*(θ)n║
Comparison between ϕ and ϕ = max _{0≤n≤100}║M_{*}(θ)^{n}║
IV. STABILITY OF METHODS OF ORDER 3 AND ORDER 4
Method (5) for
s
= 3 corresponding to the parameter values
is called Ruth’s method, which is of the order 3 in accuracy. For Ruth’s method, we have
The stability interval is
≈2.50748 where
denotes a root of tr
M
_{*}
(
θ
) = −2.
To try to improve Ruth’s method with respect to stability, we consider (5) for
s
= 4 with
, which is reduced to
At first glance, it appears that (29) needs more evaluation of
f
than (5) with
s
= 3, but
f
(
p
_{n+1}
) for the computation of
q
_{n+1}
is again used for the computation of
Q
_{1}
at the next step
t
=
t
_{n+1}
. Hence, from the perspective of function evaluation, the work needed for (29) is the same as that for (5) with
s
= 3 e. g., Ruth’s method. This idea is called first same as last and is often utilized in the numerical analysis of differential equations
[2]
.
Method (29) is of the order 3 if the parameters satisfy
These are too complicated to treat. We thus introduce the simplifying condition
By virtue of this condition, the coefficient of
θ
^{6}
in tr
M
_{*}
(
θ
) becomes 0, and the trace is reduced to
The stability interval becomes
which is larger than that of Ruth’s method.
Eqs. (30) and (31) form a system of 6 equations with 7 unknown variables, which has solutions with a free parameter, e.g.,
b
_{1}
. Letting
b
_{1}
= 1/3, we obtain the following :
We refer to the corresponding method as the stabilized 3rdorder method. In
Fig. 2
, the functions
ϕ
for Ruth’s method and the stabilized 3rdorder method are presented. For
θ
≤ 2.37,
ϕ
for Ruth’s method is smaller than
ϕ
for the stabilized 3rdorder method, but the latter has finite values up to
Several symplectic methods of the order 4 are known. Among them, a method of the form (29) corresponding to the parameter values (see, e.g.,
[4]
, p. 109)
For this method, we have the following:
The stability interval is
≈ 1.57340, where
is a root of tr
M
_{*}
(
θ
) = 2 . The stability interval is smaller than that of the symplectic Euler method (
Fig. 2
).
Functions ϕ for the three symplectic methods
V. NUMERICAL ILLUSTRATION
To test our numerical method, we consider the sine Gordon equation
This equation has the solitary wave solution (see, e.g.,
[10]
, chapter 17).
By introducing a new variable
v = 𝜕u/𝜕t
and restricting the space variable
x
to 5 ≤
x
≤ 5, we get the problem
where
φ
_{0}
(
t
) and
φ
_{1}
(
t
) are given so that (38) satisfies (39). Moreover, we apply the method of lines approximation to problem (39) by using a mesh of the form
x_{j}
= 5 +
jΔ x
,
j
= 0,1 …,
M
,
Δx
= 10/
M
, enotes a positive integer. As usual, we denote approximate functions to
u
(
t,x_{j}
) and
v
(
t,x_{j}
) by
u
_{j}
(
t
) and
v
_{j}
(
t
), respectively. By approximating 𝜕
^{2}
u
/𝜕
x
^{2}
with the standard central difference scheme, we get a Hamiltonian equation
where
q
(
t
) = [
u
_{1}
(
t
),
u
_{2}
(
t
), …,
u
_{M1}
(
t
)]
^{T}
,
p
(
t
) = [
v
_{1}
(
t
),
v
_{2}
(
t
), …,
v
_{M1}
(
t
)]
^{T}
The matrix
L
_{Δx}
has eigenvalues represented as
By using a linear transform, we change the linear part of (40) into equations of the form
Since
ω
_{M1}
is the largest among
ω_{j}
’s, a symplectic method is stable for the linear part of (40) if
ω
_{M1}
Δ
t
is included in the interior of the stability interval. Denoting the stability interval by [0,
θ
_{0}
] we express this condition as
which gives,
M
→ ∞ a CFL condition
We now consider time step sizes of the form Δ
t
= 10/
N
, where
N
is a positive integer, and assume 3
N
= 2
M
for
M
and
N
. Then, since Δ
t
/ Δ
x
= 3/2, among the specific symplectic methods in Sections 2 and 3, only the stabilized 3rdorder method satisfies the CFL condition (46).
Table 2
shows the errors
for
M
= 150, 300, 600, 1200, in the case γ = 1 ⁄2 . It is observed that the numerical solution converges to the exact solution (38) with
O
(Δ
x
^{2}
). For this selection of Δ
x
and Δ
t
, the other methods bring no significant numerical results because of overflow.
Numerical results by the stabilized 3rdorder method
Numerical results by the stabilized 3rdorder method
Acknowledgements
The authors would like to thank Ms. Wakana Tamaru, a student of Nanzan University for her help with checking the mathematical expressions and numerical results presented in the paper.
BIO
Toshiyuki Koto received his B.S. degree in 1984 and M.S. degree in 1986 from Department of Mathematics, the University of Tokyo. In 1992, he received his Ph.D. in Engineering from Nagoya University. From April 1986 to March 1991, he was with Fujitsu Limited; from April 1991 to March 2004, with the University of Electro Communications; and from April 2004 to March 2009, with Nagoya University. Since April 2009, he has been a professor at Nanzan University. His research interests include numerical analysis and applied mathematics. He is a member of the Mathematical Society of Japan, the Society of Information Processing, and the Japan Society of Industrial and Applied Mathematics.
EunJee Song received her B.S. degree from the Department of Mathematics, Sookmyung Women’s University, in 1984. She earned her M.S. and Ph.D. degrees from the Department of Information Engineering, Nagoya University, Japan in 1988 and 1991, respectively. She was an exchange professor at the Department of Computer Science, the University of Auckland, New Zealand, in 2007. She is currently a full and tenured professor of the Department of Computer Science, Namseoul University, Cheonan, Korea.
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