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3 An Extended Discrete Gradient Formula Based on ERKN Integrators

3 An Extended Discrete Gradient Formula Based on ERKN Integrators

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5.3 An Extended Discrete Gradient Formula Based on ERKN Integrators



101



Using the symmetry and commutativity of M and all φl (K) and inserting (5.18) into

(5.19), with a tedious and careful computation, we obtain

1

1

p φ 2 (K) + Kφ12 (K) pn + qn M φ02 (K) + Kφ12 (K) qn

2 n 0

2

+ qn K φ1 (K)2 − φ0 (K)φ2 (K) ∇U(qn , qn+1 )



H(pn+1 , qn+1 ) =



− hpn φ0 (K)φ1 (K) + Kφ1 (K)φ2 (K) ∇U(qn , qn+1 )

1

+ h2 ∇U(qn , qn+1 ) φ1 (K)2 + Kφ2 (K)2 ∇U(qn , qn+1 ) + U(qn+1 ).

2



(5.20)



From the definition of φi (K), it can be shown that

φ02 (K) + Kφ12 (K) = I, K φ1 (K)2 − φ0 (K)φ2 (K) = I − φ0 (K),

φ1 (K)2 + Kφ2 (K)2 = 2φ2 (K), φ0 (K) + Kφ2 (K) = I,



(5.21)



where I is the d × d identity matrix.

Substituting (5.21) into (5.20) gives

1

1

p pn + qn Mqn

2 n

2

+ qn I − φ0 (K) ∇U(qn , qn+1 ) − hpn φ1 (K)∇U(qn , qn+1 )



H(pn+1 , qn+1 ) =



+ h2 ∇U(qn , qn+1 ) φ2 (K)∇U(qn , qn+1 ) + U(qn+1 ).

(5.22)

With the definition of discrete gradient and the first equation of (5.18), (5.22) becomes

H(pn+1 , qn+1 ) =



1

1

p pn + qn Mqn + qn − φ0 (K)qn + hφ1 (K)pn

2 n

2



− h2 φ2 (K)∇U(qn , qn+1 )



∇U(qn , qn+1 ) + U(qn+1 )



1

1

p pn + qn Mqn + qn − qn+1 ∇U(qn , qn+1 ) + U(qn+1 )

2 n

2

1

1

= pn pn + qn Mqn + U(qn )

2

2

= H(pn , qn ).

(5.23)

The proof is complete.

=



Remark 5.1 It is noted that in the particular case of M = 0, the d × d zero matrix,

namely, H(p, q) = 21 p p + U(q), and the formula (5.18) reduces to the traditional

discrete gradient method (5.13). In fact, the choice of M = 0 in (5.18) gives

1

qn+1 = qn + hpn − h2 ∇U(qn , qn+1 ), pn+1 = pn − h∇U(qn , qn+1 ),

2



(5.24)



102



5 An Extended Discrete Gradient Formula …



or equivalently,

1

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

2



(5.25)



which is exactly the same as (5.13).

In what follows, we go further in studying the extended discrete gradient formula

(5.18) and give some other properties related to the formula.

First, we consider the classical algebraic order of (5.18). An integration formula

has order r, if for any smooth problem under consideration, the local truncation errors

of the formula satisfy

en+1 := q(tn+1 ) − qn+1 = O(hr+1 ) and en+1 := p(tn+1 ) − pn+1 = O(hr+1 ),

where q(tn+1 ) and p(tn+1 ) denote the values of the exact solution of the problem

and its first derivative at tn+1 = tn + h, respectively, and qn+1 and pn+1 express

the one-step numerical results obtained by the formula under the local assumptions

qn = q(tn ) and pn = p(tn ).

Theorem 5.2 Assume that ∇U(q, q ) is an approximation to the gradient of U(q) of

at least first-order at the midpoint of the interval [q, q ]. Then, the extended discrete

gradient formula (5.18) is of order two.

Proof By (5.16) and (5.18), we have

1



q(tn+1 ) − qn+1 = h2



−(1 − z)φ1 ((1 − z)2 K)∇U(q(tn + hz))dz



0



(5.26)



+ h2 φ2 (K)∇U(qn , qn+1 ).

The first equation of (5.18) gives

qn+1 − qn = (φ0 (K) − I)qn + hφ1 (K)pn − h2 φ2 (K)∇U(qn , qn+1 ).

From φ0 (K) − I = O(h2 ), it follows that

qn+1 − qn = O(h).



(5.27)



Under the local assumption qn = q(tn ), we have

q(tn + hz) − qn = O(h), 0



z



1.



(5.28)



From (5.27) and (5.28), we obtain

q(tn + hz) −



qn + qn+1

= O(h), 0

2



z



1.



(5.29)



5.3 An Extended Discrete Gradient Formula Based on ERKN Integrators



103



Thus, with (5.29) and the assumption of the theorem, (5.26) becomes

q(tn+1 ) − qn+1 = h2



1

0



−(1 − z)φ1 (1 − z)2 K



+ φ2 (K) ∇U(

= h2



1

0



∇U



qn + qn+1

+ O(h)

2



qn + qn+1

) + O(h) dz

2



− (1 − z)φ1 ((1 − z)2 K) + φ2 (K) ∇U



qn + qn+1

+ O(h)dz

2



= O(h3 ).



Similarly, we obtain p(tn+1 ) − pn+1 = O(h3 ).

The symmetry of a method is also very important in long-term integration. The

definition of symmetry is given below (see [10]).

Definition 5.2 The adjoint method Φh∗ of a method Φh is defined as the inverse map

−1

. A method with

of the original method with reversed time step −h, i.e. Φh∗ := Φ−h



Φh = Φh is called symmetric.

With the definition of symmetry, a method is symmetric if exchanging n + 1 ↔

n, h ↔ −h leaves the method unaltered.

In what follows, we show a result on symmetry for the extended discrete gradient

formula (5.18).

Theorem 5.3 The extended discrete gradient formula (5.18) is symmetric provided

∇U in (5.18) satisfies the assumption: ∇U(q, q ) = ∇U(q , q) for all q and q .

Proof Exchanging qn+1 ↔ qn , pn+1 ↔ pn and replacing h by −h in (5.18) give

qn = φ0 (K)qn+1 − hφ1 (K)pn+1 − h2 φ2 (K)∇U(qn+1 , qn ),

pn = hMφ1 (K)qn+1 + φ0 (K)pn+1 + hφ1 (K)∇U(qn+1 , qn ).



(5.30)



Multiplying both sides of the two equations in (5.30) by φ0 (K), φ1 (K), respectively,

and with some manipulation, we obtain

qn+1 = φ0 (K)qn + hφ1 (K)pn − h2 φ12 (K) − φ0 (K)φ2 (K) ∇U(qn+1 , qn ),

pn+1 = −hMφ1 (K)qn + φ0 (K)pn − h (Kφ1 (K)φ2 (K) + φ0 (K)φ1 (K)) ∇U(qn+1 , qn ).



(5.31)

Since

φ12 (K) − φ0 (K)φ2 (K) = φ2 (K)

and

Kφ1 (K)φ2 (K) + φ0 (K)φ1 (K) = φ1 (K),



104



5 An Extended Discrete Gradient Formula …



we obtain

qn+1 = φ0 (K)qn + hφ1 (K)pn − h2 φ2 (K)∇U(qn+1 , qn ),

pn+1 = −hMφ1 (K)qn + φ0 (K)pn − hφ1 (K)∇U(qn+1 , qn ),



(5.32)



which shows that the formula (5.18) is symmetric under the stated assumption.

Remark 5.2 It should be noted that all the three discrete gradients ∇ 1 U, ∇ 2 U and

∇ 3 U satisfy the assumption of Theorem 5.3.



5.4 Convergence of the Fixed-Point Iteration

for the Implicit Scheme

The previous section derived the extended discrete gradient formula (5.18) and presented some of its properties. In this section, using the discrete gradients given in

Sect. 5.2, we propose three practical schemes based on the extended discrete gradient

formula (5.18) for the Hamiltonian system (5.4):

• MVDS (mean value discrete gradient):





2





⎨ qn+1 = φ0 (K)qn + hφ1 (K)pn − h φ2 (K)



1



∇U((1 − τ )qn + τ qn+1 )dτ,



0









⎩ pn+1 = −hMφ1 (K)qn + φ0 (K)pn − hφ1 (K)



1



∇U((1 − τ )qn + τ qn+1 )dτ.



0



(5.33)



• MDS (midpoint discrete gradient):



1





qn+1 = φ0 (K)qn + hφ1 (K)pn − h2 φ2 (K) ∇U (qn + qn+1 )





2











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





(qn+1 − qn ) ,

+





|q

− q |2

n+1



n



1







pn+1 = −hMφ1 (K)qn + φ0 (K)pn − hφ1 (K) ∇U (qn + qn+1 )





2











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





(qn+1 − qn ) .

+



|qn+1 − qn |2



(5.34)



• CIDS (coordinate increment discrete gradient):

qn+1 = φ0 (K)qn + hφ1 (K)pn − h2 φ2 (K)∇U(qn , qn+1 ),

pn+1 = −hMφ1 (K)qn + φ0 (K)pn − hφ1 (K)∇U(qn , qn+1 ),



(5.35)



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:



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