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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

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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

∇ 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

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

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,

is dependent on M .

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

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

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