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
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].
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