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4 Convergence of the Fixed-Point Iteration for the Implicit Scheme

4 Convergence of the Fixed-Point Iteration for the Implicit Scheme

Tải bản đầy đủ - 0trang

5.4 Convergence of the Fixed-Point Iteration for the Implicit Scheme



105



with





1 ,q2 ,q3 ,...,qd )−U(q1 ,q2 ,q3 ,...,qd )

U(qn+1

n n

n

n n n

n

1 −q1

qn+1

n

1 ,q2 ,q3 ,...,qd )−U(q1 ,q2 ,q3 ,...,qd )

U(qn+1

n

n

n+1 n

n+1 n n

2 −q2

qn+1

n



































..

⎟,

∇U(qn , qn+1 ) := ⎜

.









d−2 d−1 d

d−2 d−1 d

1

1

⎜ U(qn+1 ,...,qn+1 ,qn+1 ,qn )−U(qn+1 ,...,qn+1 ,qn ,qn ) ⎟





d−1





qn+1

−qnd−1





1 ,...,qd−2 ,qd−1 ,qd )−U(q1 ,...,qd−2 ,qd−1 ,qd ) ⎠

⎝ U(qn+1

n+1

n+1 n+1 n+1

n+1 n+1 n

d −qd

qn+1

n



(5.36)



where qi is the ith component of q.

When M = 0, the above three methods reduce to





h2 1





q

=

q

+

hp



∇V ((1 − τ )qn + τ qn+1 )dτ,

n+1

n

n



2 0

1







∇V ((1 − τ )qn + τ qn+1 )dτ,

⎩ pn+1 = pn − h



(5.37)



0





1

h2





q

∇V ( (qn + qn+1 ))

=

q

+

hp



n+1

n

n





2

2









V (qn+1 ) − V (qn ) − ∇V ( 21 (qn + qn+1 )) · (qn+1 − qn )







+

(qn+1 − qn ) ,



|qn+1 − qn |2



1





pn+1 = pn − h ∇V ( (qn + qn+1 ))





2











V (qn+1 ) − V (qn ) − ∇V ( 21 (qn + qn+1 )) · (qn+1 − qn )





+

(qn+1 − qn ) ,

|qn+1 − qn |2

(5.38)

and



h2

⎨q

∇V (qn , qn+1 ),

n+1 = qn + hpn −

(5.39)

2



pn+1 = pn − h∇V (qn , qn+1 ),

1

respectively, where V (q) = q Mq + U(q) and ∇V (qn , qn+1 ) is the same as (5.36).

2

1

Substituting V (q) = q Mq + U(q) into (5.37), (5.38) and (5.39), respectively,

2

yields the following three schemes:



106



5 An Extended Discrete Gradient Formula …



• MVDS0:



h2





⎪ qn+1 = qn + hpn −





2









⎩ pn+1 = pn − h



1

1

M(qn + qn+1 ) +

∇U((1 − τ )qn + τ qn+1 )dτ ,

2

0



1

1

∇U((1 − τ )qn + τ qn+1 )dτ .

M(qn + qn+1 ) +

2

0



(5.40)

• MDS0:





h2 1

1







qn+1 = qn + hpn −

M(qn + qn+1 ) + ∇U (qn + qn+1 )





2

2

2











U(qn+1 ) − U(qn ) − ∇U( 21 (qn + qn+1 )) · (qn+1 − qn )





+

(qn+1 − qn ) ,



|qn+1 − qn |2



1

1





pn+1 = pn − h M(qn + qn+1 ) + ∇U (qn + qn+1 )





2

2











U(qn+1 ) − U(qn ) − ∇U( 21 (qn + qn+1 )) · (qn+1 − qn )





+

(qn+1 − qn ) .



|qn+1 − qn |2



(5.41)



• CIDS0:



h2 1





⎨ qn+1 = qn + hpn − 2 2 M(qn + qn+1 ) + ∇U(qn , qn+1 ) ,



1



⎩ pn+1 = pn − h

M(qn + qn+1 ) + ∇U(qn , qn+1 ) ,

2



(5.42)



where ∇U(qn , qn+1 ) is defined by (5.36).

The integrals in (5.33) and (5.40) can be approximated by quadrature formulae,

in such a way that we can obtain practical numerical schemes. We note that these

schemes are all implicit and require iterative solution, in general. In this chapter, we

use the well-known fixed-point iteration for these implicit schemes.

In what follows, we analyse the convergence of the fixed-point iteration for these

formulae. First, we consider the one-dimensional case.

In the one-dimensional case, the Hamiltonian system (5.4) reduces to



dU



⎨ q˙ = p, p˙ = −ω2 q −

, ω > 0, t ∈ [t0 , T ],

dq

(5.43)





q(t0 ) = q0 , p(t0 ) = p0 ,

and the Hamiltonian reduces to

H(p, q) =



1 2 1 2 2

p + ω q + U(q).

2

2



5.4 Convergence of the Fixed-Point Iteration for the Implicit Scheme



107



The φ-functions φ0 (K), φ1 (K) and φ2 (K) in the formulae reduce to cos(ν), sinc(ν)

sin(ξ )

ν

1

, respectively, where ν = hω and sinc(ξ ) =

and sinc2

. In this case, all

2

2

ξ

the three new energy-preserving schemes and the three corresponding original ones

given in this section for (5.4) reduce to the following two schemes:



h2



2 ν U(qn+1 ) − U(qn )



,

⎨ qn+1 = cos(ν)qn + h sinc(ν)pn − sinc ( )

2

2

qn+1 − qn

(5.44)



U(qn+1 ) − U(qn )



⎩ pn+1 = −h sin(ν)qn + cos(ν)pn − h sinc(ν)

,

qn+1 − qn



h2 1 2

U(qn+1 ) − U(qn )





,

⎨ qn+1 = qn + hpn − 2 2 ω (qn+1 + qn ) +

qn+1 − qn



1 2

U(qn+1 ) − U(qn )

⎪p



.

ω (qn+1 + qn ) +

n+1 = pn − h

2

qn+1 − qn



(5.45)



We assume that U(q) is twice continuously differentiable. For fixed h, qn , pn , the

first equation of (5.44) is a nonlinear equation with respect to qn+1 . Let

F(q) = cos(ν)qn + h sinc(ν)pn −



ν U(q) − U(qn )

h2

sinc2

.

2

2

q − qn



Then qn+1 is a fixed-point of F(q) and we have

F(q) − F(q ) =

=

=



ν

U(q ) − U(qn )

h2

U(q) − U(qn )



sinc2 ( )

2

2

q − qn

q − qn

ν

h2

sinc2 ( )

2

2

h2

ν

sinc2 ( )

2

2



1



U ((1 − τ )qn + τ q) − U ((1 − τ )qn + τ q )dτ



0

1

0



1



U (1 − τ )qn + (1 − s)τ q + sτ q τ (q − q )dsdτ



0



h2

max U (ξ ) q − q .

4 ξ



By the fixed-point theorem, if



h2

4



(5.46)



max U (ξ ) < c < 1, the fixed-point iteration for

ξ



the first equation with respect to qn+1 in (5.44) is convergent. The important point

here is that the convergence of fixed-point iteration for (5.44) is independent of the

frequency ω. Similarly, for the first equation in (5.45), let

G(q) = qn + hpn −



h2

2



1 2

U(q) − U(qn )

.

ω (q + qn ) +

2

q − qn



108



5 An Extended Discrete Gradient Formula …



Then

G(q) − G(q )

=



h2 1 2

U(q) − U(qn ) U(q ) − U(qn )

ω (q − q ) +



2 2

q − qn

q − qn



=



1

h2 1 2

U ((1 − τ )qn + τ q) − U ((1 − τ )qn + τ q )dτ

ω (q − q ) +

2 2

0



=



1 1

h2 1 2

ω (q − q ) +

U (1 − τ )qn + (1 − s)τ q + sτ q τ (q − q )dsdτ

2 2

0 0



=



1 1

h2 1 2

ω +

U (1 − τ )qn + (1 − s)τ q + sτ q τ dsdτ q − q .

2 2

0 0



h2

4



Thus, if



h2

4



ω2 + max U (ξ )



q−q .



ξ



(5.47)



< c < 1, the fixed-point iteration for the first



ω2 + max U (ξ )

ξ



equation with respect to qn+1 in the scheme (5.45) is convergent. We note that the

convergence of the fixed-point iteration for (5.45) is dependent on the frequency ω.

For the multidimensional case, we only consider (5.33) and (5.40). The analysis

for the other cases is similar.

In what follows, we use the Euclidean norm and its induced matrix norm (spectral

norm) and denote them by · . Similarly to the one-dimensional case, let the righthand side of the first equation in (5.33) be

1



R(q) = φ0 (K)qn + hφ1 (K)pn − h2 φ2 (K)



∇U (1 − τ )qn + τ q dτ,



0



and in (5.40) let the right-hand side of the first equation be

I(q) = qn + hpn −



h2

2



1

M(qn + qn+1 ) +

2



1



∇U (1 − τ )qn + τ qn+1 dτ .



0



Then,

R(q) − R(q ) = h2 φ2 (K)

= h2 φ2 (K)



1

0



U ((1 − τ )qn + τ q) − U (1 − τ )qn + τ q dτ



1

0



1

0



U (1 − τ )qn + (1 − s)τ q + sτ q τ (q − q )dsdτ



h2

φ2 (K) max ∇ 2 U(ξ )

2

ξ

h2

max ∇ 2 U(ξ )

4 ξ



q−q ,



q−q



(5.48)



5.4 Convergence of the Fixed-Point Iteration for the Implicit Scheme



109



where the last inequality is due to the symmetry of M and the definition of φ2 (K).

Likewise, we have

I(q) − I(q )

=

=



h2

2



1

M(q − q ) +

2



h2

2



1

M(q − q ) +

2



h2

2



1

M+

2



h2

4



1

0



1



1



U (1 − τ )qn + τ q − U (1 − τ )qn + τ q dτ



0

1

0



1



U (1 − τ )qn + (1 − s)τ q + sτ q τ (q − q )dsdτ



0



U (1 − τ )qn + (1 − s)τ q + sτ q τ dsdτ



q−q .



0



M + max ∇ 2 U(ξ )

ξ



q−q .



(5.49)



Compared (5.48) with (5.49), it can be observed that the fixed-point iteration of

extended discrete gradient schemes has a larger convergence domain than that of

M , where

traditional discrete gradient schemes, especially for the case ∇ 2 U

∇ 2 U is the Hessian matrix of U. Moreover, it is clear from (5.49) that the convergence

of fixed-point iteration for traditional discrete gradient methods depends on M and

the larger M is, the smaller the stepsize is required to be. Whereas, it is important to

see from (5.48) that the convergence of fixed-point iteration for the implicit schemes

based on the extended discrete gradient formula (5.18) is independent of M .



5.5 Numerical Experiments

Below we apply the methods presented in this chapter to the following three problems

and show their remarkable efficiency.

Problem 5.1 Consider the Duffing equation

qă + 2 q = k 2 (2q3 − q), t ∈ [0, tend ],

q(0) = 0, q˙ (0) = ω,

with 0 ≤ k < ω. The problem is a Hamiltonian system with the Hamiltonian

H(p, q) =



k2

1 2 1 2 2

p + ω q + U(q), U(q) = (q2 − q4 ).

2

2

2



The analytic solution of this initial value problem is given by

q(t) = sn(ωt, k/ω),



110



5 An Extended Discrete Gradient Formula …



and represents a periodic motion in terms of the Jacobian elliptic function sn. In this

test, we choose the parameter values k = 0.03, tend = 10000.

Since it is a one-dimensional problem, only the methods CIDS and CIDS0 are

used. First, for the fixed-point iteration, we set the maximum iteration number as 10

and the error tolerance as 10−15 . We plot the logarithm of the errors of the Hamiltonian

EH = max |H(p5000i , q5000i )−H(p0 , q0 )| against N = 5000i for ω = 10 and ω = 20

with h = 1/24 . The results are shown in Fig. 5.1. We then set the maximum iteration

number as 1000 and the error tolerance as 10−15 and plot the logarithm of the error

tolerance of the iteration against the logarithm of the total iteration number for the

two methods with h = 1/24 . The results are shown in Fig. 5.2.

Problem 5.2 Consider the Fermi-Pasta-Ulam problem.

This problem has been considered in Sect. 2.4.4 of Chap. 2.

Following [10], we choose

m = 3, q1 (0) = 1, p1 (0) = 1, q4 (0) =



1

, p4 (0) = 1,

ω



(5.50)



and choose zero for the remaining initial values. We apply the methods MDS, MDS0,

CIDS and CIDS0 to the system. The system with the initial values (5.50) is integrated

on the interval [0, 1000] with the stepsize h = 1/100 for ω = 100 and ω = 200.

First, for fixed-point iteration, we set the maximum iteration number as 10, the

error tolerance as 10−15 and plot the logarithm of the errors of the Hamiltonian

EH = max |H(p5000i , q5000i ) − H(p0 , q0 )| against N = 5000i, i = 1, 2, . . .. If the

error is very large, we do not plot the points in the figure. The results are shown in

Fig. 5.3. As is seen from the result, for fixed h = 1/100, as ω increases from 100 to



(a)



(b)



Duffing equation: energy consevation



4



−6.0



2

0



−7.0



log10(EH)



10



(EH)



−6.5



log



Duffing equation: energy consevation



−5.5



−7.5

−8.0

−8.5

−9.0



−4

−6

−8



−9.5

−10.0

−10.5



−2



0



0.5



1.0

N=5000i, i=1,2,...



1.5



CIDS

CIDS0

2.0

5

x 10



−10

−12



0



0.5



1.0

N=5000i, i=1,2,...



1.5



CIDS

CIDS0

2.0

5

x 10



Fig. 5.1 Problem 5.1 (maximum iteration number = 10): the logarithm of the errors of Hamiltonian EH = |H(p5000i , q5000i ) − Hp0 ,q0 | against N = 5000i, i = 1, 2, . . .. a ω = 10, h = 214 .

b ω = 20, h = 214



5.5 Numerical Experiments



(b)



Duffing equation: efficiency curve



Duffing equation: efficiency curve



−10.0



−10.0



−10.5



−10.5



log10 ( error tolorence)



log



10



( error tolorence)



(a)



111



−11.0



−11.0



−11.5



−11.5



−12.0



−12.0



−12.5



−12.5



−13.0



−13.0



−13.5



−13.5



−14.0



−14.0

−14.5

−15.0

5.4



CIDS

CIDS0



CIDS

CIDS0



−14.5



5.6



5.8



6.0



6.2



6.4



−15.0

5.5



6.0



log10 (total iteration number)



6.5



7.0



log10 (total iteration number)



Fig. 5.2 Problem 5.1 (maximum iteration number = 1000): the logarithm of the error tolerance of

the iteration against the logarithm of the total iteration number. a ω = 10, h = 214 . b ω = 20, h = 214



(a)



(b)



FPU: energy consevation



−2



FPU: energy consevation

−2

−4



−6



(EH)



−10



−8



−10



CIDS

MDS

CIDS0

MDS0



−12

−14



−6



10



−8



log



log 10 (EH)



−4



0



2



4



6



8



N=5000i, i=1,2,...



10



12

4

x 10



−12

−14



0



2



4



6



8



N=5000i, i=1,2,...



CIDS

MDS

CIDS0

MDS0

10

12

4



x 10



Fig. 5.3 Problem 5.2 (maximum iteration number = 10): the logarithm of the errors of Hamiltonian

1

EH = |H(p5000i , q5000i ) − Hp0 ,q0 | against N = 5000i, i = 1, 2, . . .. a ω = 100, h = 100

.

1

b ω = 200, h = 100



200, the traditional discrete gradient methods are not convergent any more. However,

the new schemes still converge well.

Next, we set the maximum iteration number as 1000, and choose h = 1/100 for

ω = 100, h = 1/200 for ω = 200. We plot the logarithm of the error tolerance of

the iteration against the logarithm of the total iteration number for the four methods.

The results are shown in Fig. 5.4.

Problem 5.3 Consider the sine-Gordon equation with periodic boundary conditions

(see [12])



⎨ ∂ 2u

∂ 2u

= 2 − sin u, −1 < x < 1, t > 0,

2

∂t

∂x

⎩ u(−1,

t) = u(1, t).



112



5 An Extended Discrete Gradient Formula …



−8.5

−9.0

−9.5

−10.0

−10.5

−11.0

−11.5

−12.0

5.0



5.5



6.0



6.5



10



CIDS

MDS

CIDS0

MDS0



log



10



log



(b)



FPU: efficiency curve



−8.0



(iteration tolorence error)



(iteration tolorence error)



(a)



−8.0



FPU: efficiency curve



−8.5

−9.0

−9.5

−10.0

−10.5

−11.0



CIDS

MDS

CIDS0

MDS0



−11.5

−12.0



7.0



5.0



5.5



6.0



6.5



7.0



log 10 (iteration number)



log 10 (iteration number)



Fig. 5.4 Problem 5.2 (maximum iteration number = 1000): the logarithm of the error tolerance

1

of the iteration against the logarithm of the total iteration number. a ω = 100, h = 100

.bω =

1

200, h = 200



Using semi-discretization on the spatial variable with second-order symmetric

differences, and introducing generalized momenta p = q˙ , we obtain the Hamiltonian

system with the Hamiltonian

H(p, q) =



1

1

p p + q Mq + U(q),

2

2



where q(t) = u1 (t), . . . , ud (t) and U(q) = − cos(u1 ) + · · · + cos(ud ) with

ui (t) ≈ u(xi , t), xi = −1 + iΔx, i = 1, . . . , d, Δx = 2/d, and





2

⎜ −1

1 ⎜



M=



2

Δx ⎜



−1





−1

−1



2 −1





.. .. ..

⎟.

. . .



−1 2 −1 ⎠

−1 2



(5.51)



We take the initial conditions as

q(0) = π



d

,

i=1



p(0) =





2π i

d 0.01 + sin

d



d



.

i=1



The system is integrated on the interval [0, 100] with the methods MDS, MDS0,

CIDS and CIDS0. First, for fixed-point iteration, we set the maximum iteration

number as 10 and the error tolerance as 10−15 . We plot the logarithm of the errors

of the Hamiltonian EH = max |H(p100i , q100i ) − H(p0 , q0 )| against N = 100i for

d = 36 and d = 144 with h = 0.04. If the error is very large, we do not plot the

points in the figure. The results are shown in Fig. 5.5.



5.5 Numerical Experiments



113



We then set the maximum iteration number as 1000 and the error tolerance as

10−15 . We apply the four methods to the system on the interval [0, 10] with the

stepsize h = 0.04 for d = 36 and h = 0.01 for d = 144. We plot the logarithm of the

error tolerance of the iteration against the logarithm of the total iteration number for

the four methods. The results are shown in Fig. 5.6. It can be observed from the results

that compared with traditional discrete gradient methods, the fixed-point iterations

of the new methods have much larger convergence domains, and the convergence is

faster.



(a)



(b)



sine-Gordon: energy conservation



−2



sine-Gordon: energy conservation



−2

−3

−4



−6



−5



log 10 (EH)



log 10 (EH)



−4



−8



−10



−6

−7

−8

−9



CIDS

MDS

CIDS0

MDS0



−12

−14

0



500



1000



1500



2000



2500



CIDS

MDS

CIDS0

MDS0



−10

−11

−12



3000



0



500



1000



1500



2000



2500



3000



N=100i, i=1,2,...



N=100 i, i=1,2,...



Fig. 5.5 Problem 5.3 (maximum iteration number = 10): the logarithm of the errors of Hamiltonian EH = |H(p100i , q100i ) − Hp0 ,q0 | against N = 100i, i = 1, 2, . . .. a d = 36, h = 0.04.

b d = 144, h = 0.04



(b)



sine-Gordon: efficiency curve

CIDS

MDS

CIDS0

MDS0



−10.5

−11.0

−11.5



sine-Gordon: efficiency curve

−10.0

−11.0

−11.5



−12.0



−12.0



−12.5



−12.5



−13.0



−13.0



−13.5



−13.5



−14.0



−14.0



−14.5



−14.5



−15.0

2.5



CIDS

MDS

CIDS0

MDS0



−10.5



log 10 (error tolorence)



log



10



(error tolorence)



(a)−10.0



−15.0

3.0



log



3.5

10



4.0



4.5



5.0



(total iteration number)



5.5



3.0



3.5



log



4.0

10



4.5



5.0



5.5



6.0



(total iteration number)



Fig. 5.6 Problem 5.3 (maximum iteration number = 1000): the logarithm of the error tolerance of

the iteration against the logarithm of the total iteration number. a d = 36, h = 0.04. b d = 144,

h = 0.01



114



5 An Extended Discrete Gradient Formula …



5.6 Conclusions

In this chapter, combining the idea of the discrete gradient method with the

ERKN integrator, we discussed an extended discrete gradient formula for the multifrequency oscillatory Hamiltonian system. Some properties of the new formula were

analysed. It is the distinguishing feature of the new formula to take advantage of

the special structure of the system introduced by the linear term Mq, so that the

extended discrete gradient methods adapt themselves to the multi-frequency oscillatory system. Since both extended discrete gradient schemes and traditional ones are

implicit, an iterative solution procedure is required. From the convergence analysis

of fixed-point iteration for implicit schemes, it can be seen that a larger stepsize

can be chosen for extended discrete gradient schemes than for traditional discrete

gradient methods, when they are applied to oscillatory Hamiltonian systems. The

convergence rate of extended discrete gradient methods is much higher than that of

traditional methods. The numerical experiments clearly support this point. Another

very important property is that the convergence rate of the fixed-point iteration for

extended discrete gradient schemes is independent of M . Unfortunately, however,

the convergence rate of fixed-point iteration for traditional discrete gradient methods

is dependent on M .

The material of this chapter is based on the work by Liu et al. [15].



References

1. Betsch P (2006) Energy-consistent numerical integration of mechanical systems with mixed

holonomic and nonholonomic constraints. Comput Methods Appl Mech Eng 195:7020–7035

2. Brugnano L, Iavernaro F, Trigiante D (2010) Hamiltonian boundary value methods (energy

preserving discrete line integral methods). J Numer Anal Ind Appl Math 5:17–37

3. Cie´sli´nski JL, Ratkiewicz B (2011) Energy-preserving numerical schemes of high accuracy for

one-dimensional hamiltonian systems. J Phys A: Math Theor 44:155206

4. Dahlby M, Owren B, Yaguchi T (2011) Preserving multiple first integrals by discrete gradients.

J Phys A: Math Theor 44:305205

5. García-Archilla B, Sanz-Serna JM, Skeel RD (1999) Long-time-step methods for oscillatory

differential equations. SIAM J Sci Comput 20:930–963

6. González AB, Martín P, Farto JM (1999) A new family of Runge-Kutta type methods for the

numerical integration of perturbed oscillators. Numer Math 82:635–646

7. Gonzalez O (1996) Time integration and discrete Hamiltonian systems. J Nonlinear Sci 6:449–

467

8. Hairer E, Lubich C (1997) The life-span of backward error analysis for numerical integrators.

Numer Math 76:441–462

9. Hairer E, Lubich C (2000) Long-time energy conservation of numerical methods for oscillatory

differential equations. SIAM J Numer Anal 38:414–441

10. Hairer E, Lubich C, Wanner G (2006) Geometric numerical integration: structure-preserving

algorithms, 2nd edn. Springer, Berlin

11. Harten A, Lax PD, van Leer B (1983) On upstream differencing and Godunov-type schemes

for hyperbolic conservation laws. SIAM Rev 25:35–61

12. Hochbruck M, Lubich C (1999) A Gautschi-type method for oscillatory second-order differential equations. Numer Math 83:403–426



References



115



13. Iavernaro F, Pace B (2007) s-stage trapezoidal methods for the conservation of Hamiltonian

functions of polynomial type. AIP Conf Proc 936:603–606

14. Itoh T, Abe K (1988) Hamiltonian conserving discrete canonical equations based on variational

difference quotients. J Comput Phys 77:85–102

15. Liu K, Shi W, Wu X (2013) An extended discrete gradient formula for oscillatory Hamiltonian

systems. J Phys A: Math Theor 46:165203

16. McLachlan RI, Quispel GRW, Robidoux N (1999) Geometric integration using discrete gradients. Phil Trans R Soc Lond A 357:1021–1045

17. McLachlan RI, Quispel GRW (2002) Splitting methods. Acta Numer 11:341–434

18. Quispel GRW, Capel HW (1996) Solving ODEs numerically while preserving a first integral.

Phys Lett A 218:223–228

19. Quispel GRW, McLaren DI (2008) A new class of energy-preserving numerical integration

methods. J Phys A: Math Theor 41:045206

20. Quispel GRW, Turner GS (1996) Discrete gradient methods for solving ODEs numerically

while preserving a first integral. J Phys A: Math Gen 29:L341–L349

21. Sanz-Serna JM (1992) Symplectic integrators for Hamiltonian problems: an overview. Acta

Numer 1:243–286

22. Wu X, You X, Li J (2009) Note on derivation of order conditions for ARKN methods for

perturbed oscillators. Comput Phys Commun 180:1545–1549

23. Wu X, You X, Shi W, Wang B (2010) ERKN integrators for systems of oscillatory second-order

differential equations. Comput Phys Commun 181:1873–1887

24. Wu X, You X, Xia J (2009) Order conditions for ARKN methods solving oscillatory systems.

Comput Phys Commun 180:2250–2257

25. Yang H, Wu X (2008) Trigonometrically-fitted ARKN methods for perturbed oscillators. Appl

Numer Math 58:1375–1395



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