Long-time averaging for integrable Hamiltonian dynamics · Long-time averaging for integrable Hamiltonian dynamics 3 where I1,...,Id are the d invariants of the problem (1), coincide

Numerische Mathematik manuscript No.(will be inserted by the editor)

Long-time averaging for integrableHamiltonian dynamics

Eric Cances1, Francois Castella2,3, Philippe Chartier3,Erwan Faou3, Claude Le Bris1, Frederic Legoll1,4, GabrielTurinici1

1 CERMICS, Ecole Nationale des Ponts et Chaussees, Marne-La-Vallee, FRANCEand MICMAC, INRIA, Rocquencourt, FRANCE.

2 IRMAR, University of Rennes I, Rennes, FRANCE.3 IPSO, INRIA, Rennes, FRANCE.4 EDF, R&D, Analyse et Modeles Numeriques, Clamart, FRANCE.

July 1, 2004

Summary Given a Hamiltonian dynamical system, we address thequestion of computing the space-average of an observable throughthe limit of its time-average. For a completely integrable system, itis known that ergodicity can be characterized by a diophantine con-dition on its frequencies and that the two averages then coincide. Inthis paper, we show that we can improve the rate of convergence uponusing a filter function in the time-averages. We then show that thisconvergence persists when a symplectic numerical scheme is appliedto the system, up to the order of the integrator.

Key words integrable Hamitonian systems – averaging – filterfunctions – Riemann sums – symplectic solvers – invariant tori

1 Introduction

Consider a Hamiltonian dynamical equation in Rd ! Rd

!p(t) = "#qH(p(t), q(t)), p(0) = p0,q(t) = #pH(p(t), q(t)), q(0) = q0.

(1)

Let M(p0, q0) be the manifold {(p, q) $ R2d |H(p, q) = H(p0, q0)}.The solution of (1) is a dynamical system on M(p0, q0) with the

2 Eric Cances et al.

invariant measure

d!(p, q) =d"(p, q)

%#H(p, q)%2

,

where d"(p, q) is the measure induced on M(p0, q0) by the Euclideanmetric of R2d (see for instance [4]), and % · %

2the Euclidean norm in

R2d.It is a common problem to estimate the space average of an ob-

servable1 A over the manifold M(p0, q0)"M(p0,q0)

A(p, q)d!(q, p)"M(p0,q0)

d!(q, p), (2)

through the limit of the time average

limT!"

1T

# T

0A(p(t), q(t))dt, (3)

where (p(t), q(t)) is the solution of (1). Our wish is here to give asound ground to (and in some cases improve [3]) the numerical sim-ulations of (3) commonly used in the field of molecular dynamics.

The conditions under which the two quantities (2) and (3) coincidecan not be stated in general. In contrast, a dynamical system is knownto have an ergodic behavior in two clearly identified -and somewhatopposite- situations:

– in the case of a di!erential equation with an hyperbolic structure,giving rise to mixing, the convergence of (3) toward (2) for T goingto infinity is insured at a typical rate of 1/

&T . It is the belief of

the authors that not much can be gained in this situation due tothe presence of chaos;

– in the case of an integrable system, a well-known result of Bohl,Sierpinski and Weyl (see [2] and references therein) states that,under a non-resonant condition on the frequency vector associ-ated with the initial condition, the space average of a continuousfunction on the manifold

S(p0, q0) = {(p, q) $ Rd ! Rd ;I1(p, q) = I1(p0, q0), . . . , Id(p, q) = Id(p0, q0)}, (4)

1 Properties of a physical system at thermodynamical equilibrium such as radialdistributions, free energies, transport coe!cients can be computed as averages ofsome observables over the phase space of a representative microscopic system.In most applications of interest, this microscopic system is composed of a highnumber of particles, making the computation of averages a challenging issue.

Long-time averaging for integrable Hamiltonian dynamics 3

where I1, . . . , Id are the d invariants of the problem (1), coincidewith the long-time average of this function. Moreover, if the fre-quencies satisfy a diophantine condition, the convergence is of or-der T#1. Being more analytically tractable, this case allows for thedesign of more elaborated averaging methods than the straight-forward numerical simulation of (3).

In realistic situations, Hamiltonian systems belong nor to the firstcategory, neither to the second one: they typically exhibit di!erentbehaviors for di!erent energy levels. Nevertheless, the accelerationtechniques presented in this paper are relevant to actual computa-tions for the following two reasons:

– their e"ciency shows o! also in situations where integrability as-sumptions are not satisfied (see the companion paper [3]).

– their induced computational overhead is only marginal and thusnot penalizing when integrability assumptions are violated. Mean-while, when the explored energy level is such that the system canbe (locally) considered as integrable, a significant acceleration isobserved.

Integrable systems under some diophantine condition will thus con-stitute a natural framework for this work. Besides, all the resultspresented here could be extended with only minor modifications tothe case of near-integrable systems.

In the following, we consider a completely integrable Hamiltoniansystem (1) in the sense of the Arnold-Liouville theorem [2,5]: Thereexist d invariants I1 = H, I2, . . . , Id in involution (i.e. their PoissonBracket {Ii, Ij} = 0) such that their gradient are everywhere inde-pendent, and the trajectories of the system remain bounded. Underthese conditions, there exist action-angles variables (a, #) in a neigh-borhood U of S(p0, q0). We have (p, q) = $(a, #), where $ is a sym-plectic transformation

$ : D ! Td ' (a, #) () (p, q) $ U,

with Td = (R/2%Z)d the standard d-dimensional flat torus, and D aneighborhood in Rd of the point a0 such that (a0, #0) = $#1(p0, q0).By definition of action-angle variables, the Hamiltonian H(p, q) of(1) is writen H(p, q) = K(a) in the coordinates (a, #), and thus thedynamics reads !

a(t) = 0,#(t) = &(a(t)), (5)


where & = 'K/'a is the frequency vector associated with the prob-lem. The solution of this system for initial data (a0, #0) is simplywriten a(t) = a0 and #(t) = &(a0)t + #0.

For fixed (a0, #0) = $(p0, q0), the image of S(p0, q0) under $#1

is the torus {a0} ! Td. On this torus, the measure d# is invariantby the flow of (5). Considering the pull-back of this measure by thetransformation $, we thus get a measure dµ(p, q) on S(p0, q0) whichis invariant by the flow of (1). For any function A(p, q) defined onS(p0, q0) we define the space average:

*A+ :=

"S(p0,q0)

A(p, q)dµ(p, q)"S(p0,q0)

dµ(p, q)=

1(2%)d

#

TdA , $(a0, #)d#. (6)

For a fixed time T , the time average is defined as

*A+(T ) :=1T

# T

0A(p(t), q(t))dt. (7)

In a first step, we will investigate the extent to which the conver-gence of the time average (7) toward the space average (6) can beaccelerated through the use of weighted integrals of the form

*A+!(T ) :=" T0 (

$tT

%A(p(t), q(t))dt

" T0 (

$tT

%dt

, (8)

where ( is a positive smooth function with compact support in [0, 1](later on, we will refer to ( as the filter function; it is sometimesrefereed as a window function in the context of signal processing[10]). In a second step, we will consider the time-discretization of(8), i.e. the discretization of both the integral through Riemann sumsand the trajectory with symplectic integrators. In particular, we willderive estimates of the convergence with respect to T and the size hof the time-grid, which are in perfect agreement with the numericalexperiments conducted in [3].

2 The complete analysis of the d-dimensional harmonicoscillator

In this section, we illustrate the main ideas of the paper in the rathersimple situation of the d-dimensional harmonic oscillator, where mostof the analysis can be conducted in an explicit way. Hereafter, H(p, q)is thus the Hamiltonian function from Rd ! Rd to R defined as

H(p, q) =12

d&

k=1

(&2kq

2k + p2

k), (9)


and the corresponding dynamics is governed by the equations!

pk = "&2kqk

qk = pk, k = 1, . . . , d.

The exact trajectory lies on the d-dimensional manifold S(p0, q0) de-fined by (4) where the Ik(p, q) = 1

2

$&2

kq2k + p2

k

%are the conserved

energies of the d oscillators. Hence, denoting r0k =

'2Ik(p0, q0),

k = 1, . . . , d and z = (&1q1 + ip1, . . . ,&dqd + ipd) the aggregatedvector of rescaled positions and momenta, the exact solution is of theform

z(t) =(r01e

i("1t+#1), . . . , r0de

i("dt+#d))

, (10)

where ) = ()1, . . . ,)d) depends on the initial conditions (p0, q0). Asa consequence, the space average (6) we wish to approximate may bewritten here as:

*A+ =1

(2%)d

#

Td(A ,*)(r0, #)d#,

where *(r0, #) = ( r01

"1cos(#1), r0

1 sin(#1), . . . ,r0d

"dcos(#d), r0

d sin(#d)). Asfor the time-average (7), it reads:

*A+(T ) =1T

# T

0(A ,*)(r0,&t + ))dt.

In order to estimate the rate of convergence of (7) toward (6), weexpand A , * in a Fourier series (the conditions under which thisexpansion is valid will be detailed in the following sections):

(A ,*)(r0, #) =&

$$Zd

!A ,*(r0,+)ei$·% ,

where + · # = +1 #1 + . . . + +d #d and with:

!A ,*(r0,+) =1

(2%)d

#

Td(A ,*)(r0, #)e#i$·%d#.

In particular, !A ,*(r0, 0) = *A+. Hence, we have:

|*A+ " *A+(T )| - 1T

&

$$Zd, $%=0

2|!A ,*(r0,+)||+ · &|

. (11)


This infinite sum can then be bounded if we assume, on one hand,that the vector of frequencies & = (&1, . . . ,&d) satisfies Siegel’s dio-phantine condition

. ,, - > 0, /+ $ Zd, |+ · &| > ,|+|#& , (12)

and on the other hand, that the Fourier coe"cients decay su"cientlyrapidly. This relatively poor rate of convergence (1/T ) may now beconsiderably improved by considering iterated averages of the form:

*A+k(T ) :=1

T k

# T

0. . .

# T

0(A ,*)(r0, (t1 + . . . + tk)& + ))dt1 . . . dtk.

(13)Using Fourier expansions as in (11), we indeed obtain in a very similarway the following error estimate for (13):

|*A+ " *A+k(T )| - 1T k

&

$$Zd, $%=0

2|!A ,*(r0,+)||+ · &|k , (14)

and under slightly more stringent bounds on the |!A ,*(r0,+)|, (14)leads to a rate of convergence of 1/T k. Inspired by these computa-tions, and noticing that (13) is a special case of (8) (more precisely*A+k(T/k) = *A+!(T ) with ( 0 .&k

[0,1/k], the kth-convolution of thecharacteristic function of [0, 1/k]), we will consider in the sequel moregeneral filter functions and demonstrate that the rate of convergencecan be further improved.

Now, a natural question that arises is whether the techniques ex-plained above are amenable to numerical computations, when boththe trajectory z(t) and the integrals (7) or (13) are approximatedusing numerical schemes. In the case of the harmonic oscillator, itturns out that the numerical trajectory zh(tn) (i.e. the approxima-tion at time tn = nh of z(tn)), when the underlying scheme is asymplectic (or symmetric) Runge-Kutta method, may be interpretedas the exact solution of a harmonic oscillator with modified frequen-cies &h

k = &k/(h&k). In particular, the numerical trajectory lies onthe same manifold S(q0, p0) as the exact one. For the velocity-Verletscheme (and partitioned methods), the same interpretation is possi-ble, though the numerical trajectory would lie on an invariant torusO(h2)-close to S(q0, p0): this situation is more typical of what hap-pens for general integrable Hamiltonian systems.

In our situation, we have:

zh(tn) =(r01e

i("1'(h"1)tn+#1), . . . , r0de

i("d'(h"d)tn+#d))

,


where / is a smooth function defined by

/(y) =1y

arctan*

R(iy) " R("iy)i(R(iy) + R("iy))

+,

R(z) being the stability function of the method (in fact, / is realanalytic as soon as R has no pole on the imaginary axis and satisfies/(y) = 1 + O(yr) where r denotes the order of convergence of theRunge-Kutta method). As a consequence, the Riemann sum associ-ated with (13) (note that (15) with k = 1 corresponds to (7)) reads,for T = nh, n $ N,

*A+Riek (T ) :=

1nk

n#1&

j1=0

. . .n#1&

jk=0

(A ,*)(r0, (j1 + . . . + jk)h& /(&h) + )),

(15)where &/(&h) = (&1/(&1h), . . . ,&d/(&dh)), so that using once againFourier expansions, we get straightforwardly:

|*A+ " *A+Riek (T )| - 1

nk

&

$$Zd, $%=0

|!A ,*(r0,+)|

,,,,,einh$·("'("h)) " 1eih$·("'("h)) " 1

,,,,,

k

.

(16)Bounding the above infinite sum now requires to bound the term|einx " 1|/|eix " 1| for x of the form x = h+ · (&/(&h)). To this aim,we use the following two inequalities

.C0, x0 > 0, /n $ N, / |x| - x0,

,,,,einx " 1eix " 1

,,,, - C01|x|

, (17)

/n $ N, /x $ R,

,,,,einx " 1eix " 1

,,,, - n, (18)

according to whether |x| is small (17) or not (18). The bound we arelooking for is now based on the following lemma:

Lemma 1 Assume that the vector of frequencies & satisfies the dio-phantine condition (12) and the Runge-Kutta method is of order r.Then, there exist strictly positive constants c and h0 such that

/h - h0 /+ $ Zd, |+ · (&/(&h))| - ,

2|+|#& =1 |+| 2 c h# r

!+1 .

Proof. Assume that there exists + $ Zd such that|+ · (&/(&h))| - ,

2|+|#& .


Then, from /(h&k) = 1 + O(|h&k|r), we obtain for h su"cientlysmall:

,

2|+|#& 2 |& · +|" C|+| |h&|r,

2 ,|+|#& " C|+| |h&|r,where C is the strictly positive constant contained in the term O(note that if / 0 1, although the constant C is zero, there is no +violating condition (12) and the lemma remains valid). Hence,

|+| 2*

,

2C|&|rh#r

+ 1!+1

.

But for |+| - ch#r/(&+1) we have |h+ · &/(&h)| - ch1#r/(&+1) for aconstant c independent of h. Hence if - > r"1, then for small enoughh we have |h+ · &/(&h)| - x0 defined in (17). Now we can split thesum in (16) into

&

1'|$|'ch! r

!+1

|!A ,*(r0,+)| Ck0

nkhk|+ · (&/(&h))|k

+&

|$|(ch! r

!+1

|!A ,*(r0,+)|. (19)

Using Lemma 1 for the first term and assuming that the Fouriercoe"cients |!A ,*(r0,+)| decay exponentially with |+|, an estimateof the form

|*A+ " *A+Riek (T )| = O

*1

T k+ exp

(" ch#s

)+,

with s = r/(- + 1). Whenever a symplectic partitioned method isused, the quadratic invariants Ik might be preserved only up to theorder of the scheme, and an additional term hr then comes into playwhich becomes dominant: for general Runge-Kutta methods, the bestpossible estimate is thus of the form

|*A+ " *A+Riek (T )| = O

*1

T k+ hr

+. (20)

The term 1/T k is the intrinsic error component of the iterated-average,whereas the term hr reflects the use of a numerical scheme of orderr. It is worth noticing that there is no secular component in hr

(neither in the bound e#c

hs ): symplectic schemes (partitioned or not)preserve quadratic invariants for all times (either exactly or up to the


order of the method). Our aim in next sections is to prove that (20)remains true over exponentially long times for averages with generalfilter functions and for general integrable Hamiltonian systems withbounded trajectories.

3 Approximation of the average: The continuous case

The function ( considered in Formula (8) is somewhat arbitrary.The most commonly used function in practice is ( 0 1, which corre-sponds to the usual time-average as defined in (7), for which conver-gence when T tends to infinity is rather slow (with rate 1/T ). For theharmonic oscillator, we have seen that the use of iterated-averages(which can be seen as a special case of filtered-averages) allows fora significant acceleration of the convergence. Theorem 1 below showsthat with increasingly smooth functions ( satisfying appropriate zeroboundary conditions, it is possible to improve the rate of convergenceto 1/T k for any integer k > 1, not only for the harmonic oscillator,but for a general integrable Hamiltonian system. It is then natural toinvestigate what happens in the limit when k tends to infinity. To thisaim, we shall consider, as an example of infinitely di!erentiable func-tions ( with compact support [0, 1] that satisfy ((k)(0) = ((k)(1) = 0for any k $ N, the function 0 defined below:

0 : [0, 1] ") [0,+3[

x (") exp*" 1

x(1 " x)

+. (21)

In the sequel, we shall assume that the estimates

%0(k)%L1 :=

# 1

0|0(k)(x)|dx - µ1kk(k, (22)

%0(k)%L" := sup

x$[0,1]|0(k)(x)| - µ1kk(k, (23)

hold for some strictly positive constants µ, 1 and 2. The existence ofsuch constants will be shown in appendix (Lemma 3).

Theorem 1 Consider the completely integrable system (1), and as-sume that the diophantine condition (12) is satisfied for &(a0) definedin (5) by the initial condition (q0, p0), with (q0, p0) = $(a0, #0). Con-sider a function A real analytic on Rd ! Rd (the observable). Recallthat to this function we associate the space-average *A+, the time-average *A+(T ) and the filtered time-average *A+!(T ) respectivelydefined in (6), (7) and (8), where ( $ C0(0, 1) (the filter function)


is assumed to be positive. Then we have the following convergenceestimates:

1. There exists a constant c depending on A, d, - and , such that

|*A+(T ) " *A+| - c

T.

2. Let k 2 1. If ( is Ck+1(0, 1) with ((j)(0) = ((j)(1) = 0 for all j =0, . . . , k"1, then there exist positive constants c0 and R dependingon A, (, d, - and ,, such that (here - $ N, though a similarformula holds for general - using the 3 function)

|*A+(T ) " *A+| - c(k,()T k+1

,

where

c(k,() = c0Rk+1(-(k + 1) + 1)!

! 1%(%

L1

(|((k)(0)| + |((k)(1)| + %((k+1)%

L1

)

3. If 0 defined in (21) is taken as the filter function, then there existstrictly positive constants c1, 4 and ! depending on A, d, - and,, such that

|*A+)(T ) " *A+| - c1e#*T 1/"

.

Proof. Statement 1. is proved in Arnold [2]. It may also be obtainedas a special case of 2. with ( 0 1. Now, if A is real analytic on Rd!Rd,then so is A ,$ on the d-dimensional torus Td and we can expand itas a Fourier series

(A , $)(a0,+) =&

$$Zd

!A , $(a0,+)ei$·% ,

with exponentially decaying coe"cients:

/+ $ Zd, |!A , $(a0,+)| - Ce#|#|C ,

where C is a strictly positive real constant. The integral over Td ofthe first coe"cient of the series (+ = 0) is straightforwardly identifiedas the space-average

!A , $(a0, 0) =1

(2%)d

#

Td(A , $)(a0, #)d#.


Writing" T0 (

$tT

%dt = T%(%

L1 := .#1, the error can be computed asfollows:

*A+!(T )"*A+ = .&

$$Zd, $%=0

!A , $(a0,+)# T

0(( t

T

)ei$·(%0+t"(a0))dt

= .&

$$Zd, $%=0

!A , $(a0,+)ei($·%0)# T

0(( t

T

)eit($·"(a0))dt. (24)

Now, the integral in each term of the series can be integrated by parts# T

0(( t

T

)eit($·"(a0))dt =

-($

tT

%eit($·"(a0))

i(+ · &(a0))

.T

0

" 1T i(+ · &(a0))

# T

0()( t

T

)eit($·"(a0))dt.

Integrating repeatedly by parts, this last term can be written as

eiT ($·"(a0))((1) " ((0)i(+ · &(a0))

" 1T i(+ · &(a0))

# T

0()( t

T

)eit($·"(a0))dt

= . . . =("1)k

(T i(+ · &(a0)))k

# T

0((k)

( t

T

)eit($·"(a0))dt,

and eventually,# T

0(( t

T

)eit($·"(a0))dt =

("1)k

(T i(+ · &(a0)))k+1T

/((k)

( t

T

)eit($·"(a0))

0T

0

" ("1)k

(T i(+ · &(a0)))k+1

# T

0((k+1)

( t

T

)eit($·"(a0))dt.

Inserting this expression in equation (24) and taking the moduli ofboth sides, we finally get the bound

|*A+!(T ) " *A+| -(|((k)(0)| + |((k)(1)| + %((k+1)%

L1)T k+1%(%

L1

!&

$$Zd, $%=0

|!A , $(a0,+)||+ · &(a0)|k+1

It remains to justify the convergence of the series considered above(and to bound its limit). This is a consequence of the diophantine


condition |+ · &(a0)| - +|$|! , which gives here

&

$$Zd, $%=0

|!A , $(a0,+)||+ · &(a0)|k+1

-&

$$Zd, $%=0

Ce#|#|C

*|+|,1/&

+&(k+1)

,

- C5&(k+1)&

$$Zd, $%=0

e#|#|C

*|+|5,1/&

+&(k+1)

.

We now take 5 = 2C+1/! so that 1/(,1/&5) = 1/(2C) and we obtain:

&

$$Zd, $%=0

|!A , $(a0,+)||+ · &(a0))|k+1

- C5&(k+1)(-(k + 1) + 1)!&

$$Zd

e#|#|2C ,

- C(2C)&(k+1)(8C)d

,k+1(-(k + 1) + 1)!,

where we have used xn - ex(n + 1)!. Statement 3. is a consequenceof Statement 2. with a suitably chosen k: since 0(k)(0) = 0(k)(1) = 0for any k $ N, we have indeed that for all k 2 0:

|*A+)(T ) " *A+| - c1

(r1

T

)k+1(k + 1)((k+1)(-(k + 1) + 1)!,

with c1 = c0µ and r1 = R1, µ and 1 being the constants of (22).Now let - be the nearest integer to - toward infinity. This gives:

|*A+)(T ) " *A+| - c1

*r1- &

T

+k+1

(k + 1)((+&)(k+1),

- c1ef(k+1),

where f(6) = 6[log(r1- &/T ) + (2 + -) log(6)]. The minimum of f for

positive 6 is attained for 6 = 1e

(T

r1&!

)1/(&+()and is

fmin = "(2 + -)e

*T

r1- &

+ 1($+!)

.

Remark 1 In the proof of Theorem 1, one gets c0 = C(8C)d, R =(2C)&/,, c1 = µc0, 4 = (2 + -)e#1-#

!!+$ and ! = (2 + -), where

- = -+1. The values of these constants rely heavily on the sharpnessof estimates (22) and it is likely that they might be improved. Nev-ertheless, the convergence behavior would be essentially the same forlarge dimensions: even if 0 were analytic, one would get ! = 1 + -.More noticeably, since almost all frequencies &(a0) satisfy the dio-phantine condition for some , as soon as - > d " 1, we may think


of - as being d and thus 2 as being approximately 1 + d. The rateof convergence thus directly depends on the dimension of the phase-space.

4 Semi-discrete averages

We now wish to investigate whether the estimates of Theorem 1 per-sist when one replaces the integrals by Riemann sums. It turns out,quite remarkably, that its proof can be almost readily adapted.

Theorem 2 Assume that the conditions of Theorem 1 are satisfiedand let T = nh > 0 for a given integer n 2 2. Let us further definethe Riemann sums corresponding to the continuous time-average

*A+Rie(T ) :=1n

n#1&

j=0

A(p(jh), q(jh)),

and the filtered time-average

*A+Rie! (T ) :=

1n#1j=0 (( j

n)A(p(jh), q(jh))1n#1

j=0 (( jn )

,

where ( $ C0(0, 1) is the filter function. Then we have the followingconvergence estimates:

1. There exist constants c and c& depending on A, d, -, , and & =&(a0) such that

,,*A+Rie(T ) " *A+,, - c

T+ c& exp

*" 1

c&h

+.

2. Let k 2 1. If ( is Ck+1(0, 1) with ((j)(0) = ((j)(1) = 0 for allj = 0, . . . , k " 1, then there exist strictly positive constants c&, c0

and R depending on A, (, d, -, , and & such that

,,*A+Rie(T ) " *A+,, - c(k,()

T k+1+ c& exp

*" 1

c&h

+,

where

c(k,() = c0Rk+1kk(-(k + 1) + 1)!

! 1%(%

L1

(|((k)(0)| + |((k)(1)| + %((k+1)%

L"

).


3. If 0 is taken as the filter function, then there exist strictly positiveconstants c&, c1, 4 and ! depending on A, d, -, , and &, such that

,,*A+Rie) (T ) " *A+

,, - c1e#*T 1/"

+ c& exp*" 1

c&h

+.

Remark 2 In the proof of Theorem 2, one gets ! = (2 + 1 + -) and4 = (2 + 1 + -)e#1(-)#

!!+$+1 , where - = - + 1.

Proof. Statement 1. is a special case of Statement 2. with ( 01, so that we focus on the error estimate for the filtered average.Expanding (A , $) in Fourier series as in Theorem 1 and denotingSn =

1n#1j=0 (1/n)((j/n), we have:

*A+Rie! (T ) " *A+ =

1nSn

&

$$Zd, $%=0

!A , $(a0,+)ei($·%0)

!n#1&

j=0

(( j

n

)ei$·jh",

where & = &(a0). We use the following result, whose proof is givenin Appendix:

Lemma 2 For a given filter-function ( in Ck+1(0, 1) with ((j)(0) =((j)(1) = 0 for all j = 0, . . . , k " 1, and a given integer n 2 k + 2,let (j be the real numbers defined by (j = ((j/n) for j = 0, . . . , n. Ifb 4= 1 is a complex number of modulus 1, then we have the estimate,,,,,,

&

0'j'n#1

(jbj

,,,,,,- 2e2kk

nk|1 " b|k+1

(|((k)(0)| + |((k)(1)| + %((k+1)%

L"

).

Now, we can bound the previous sum by using the following splitting

|*A+Rie! (T ) " *A+| - C(k,()

nk+1Sn

&

$$Zd, 0<|$|'(h|"|)!1

|!A , $(a0,+)||1 " ei$·h"|k+1

+&

$$Zd, |$|>(h|"|)!1

|!A , $(a0,+)|. (25)

where we have denotedC(k,() = 2e2kk

(|((k)(0)| + |((k)(1)| + %((k+1)%

L"

).

Note that, since 0 < |+| - (h|&|)#1 in the first term, we have0 < |h+ · &| - 1, so that b = eih$·" 4= 1.


The second term in the right-hand side can be straightforwardlybounded by c& exp(" 1

c#h). Using (12), we have for all |+| - (h|&|)#1:1

|1 " ei$·h"|- C0

1h|+ · &|

.

The first term in the right-hand side can be estimated as

C(k,()Ck+10

T k+1Sn

&

$$Zd, 0<|$|'(h|"|)!1

|!A , $(a0,+)||+ · &|k+1

and we can conclude as in the proof of Theorem 1.

5 Fully discrete averages

We consider the numerical trajectory (pn, qn) for n 2 0 obtained bya symplectic rth-order numerical scheme 7h from the initial point(p0, q0) = $#1(a0, #0).

For T = nh and n $ N, the corresponding Riemann sum reads

*A+Rie!,h(T ) :=

1n#1j=0 (( j

n )A(pj , qj)1n#1

j=0 (( jn)

. (26)

Theorem 4.4 and 4.7 of Chapter X in [5], which strongly rely on thetheory developed by Kolmogorov, Arnold and Moser [1,6–9] and onresults from the backward analysis (see [5] pp. 288 and referencestherein), yield the following result:

Theorem 3 (Hairer, Lubich, Wanner [5]) Let a& $ Td such that&(a&) satisfies the diophantine condition (12) with constants , and -,and suppose that H(p, q) is analytic on a neighborhood of the torus{(p, q) = $(a&, #) | # $ Td}. Then there exists positive constants !,c0, c, C0 and h0 such that for all h - h0 and µ - min(r,+) where+ = - + d + 1, the following holds: There exists a symplectic changeof coordinates $h : (a, #) () (b,.) analytic for

%a " a&% - c0h2µ and # $ U, = {# $ Td + iRd | |Im#| < ! }

and hr-close to the identity in the sense that

%(a, #) " $h(a, #)% - C0hr for %a " a&% - c0h

2µ and # $ U,/2,

such that in coordinates (b,.), the numerical trajectory (bn,.n) =$#1

h , $#1(pn, qn) satisfies

bn = b0 + O(exp("ch#µ/$)),

.n = nh&h(b0) + O(h#2µ/$ exp("ch#µ/$)),(27)


for nh - exp(ch#µ/$), where &h(b) = &(b) + O(hr) uniformly in b.

Using this result, we get the following Theorem:Theorem 4 Under the conditions and notations of Theorem 3, if thenumerical trajectory starts with

%a0 " a&% - c0h2µ (28)

where (a0, #0) = $#1(p0, q0), then we have:1. If ( is Ck+1 with ((j)(0) = ((j)(1) = 0 for all j = 0, . . . , k " 1

and if A is real analytic on Rd, then there exist constants c1 andC depending on A, ,, -, d, k, (, such that

/h - h0 /T = nh - exp(c1h#µ/$),

|*A+Rie!,h(T ) " *A+| - C

*1

T k+1+ hr

+. (29)

2. If 0 is taken as the filter function, if A is real analytic, then thereexist constants c1 and C, depending on A, ,, - and d such that

/h - h0 /T = nh - exp(c1h#µ/$),

|*A+Rie),h (T ) " *A+| - C

(e#*T 1/"

+ hr)

. (30)

Proof. With the notation Sn =1n#1

j=0 (1/n)((j/n), we have usingTheorem 3 that

*A+Rie!,h(T ) :=

1nSn

n#1&

j=0

((j

n)A , $ , $h(bj ,.j).

Using the Fourier expansion of A , $ , $h and (27), we obtain

*A+Rie!,h(T ) :=

1nSn

&

$$Zd

!A , $ , $h(b0,+)ei$·!0

n#1&

j=0

(( j

n

)ei$·jh"h

+ O(exp("ch#µ/$)) (31)

for nh - O(exp(ch#µ/$)) (we write c for a generic constant in theexponential), where &h = &h(b).As $h is an analytic function O(hr)-close to the identity, we have

!A , $ , $h(b0, 0) = *A+ + O(hr),

and the Fourier coe"cients !A , $ , $h(b0,+) decay exponentially withrespect to +, uniformly with respect to h. Now similarly to Lemma 1we get that

/h - h0 /+ $ Zd, |+ · (h&h))| - ,

2|+|#& =1 |+| 2 c h# r

!+1 .


And we conclude as in the proof of Theorem 2 using Lemma 2 and asplitting similar to (25).

6 Remarks on the implementation and numericalexperiments

Though optimal with respect to the rate of convergence, the filterfunction 0 does not seem to allow for the derivation of an error esti-mate: Given that the values of the constant C in (30) is out of reach,the value of n for which

R!n :=

1nj=0 ((j/n)Aj

n%(%L1

becomes su"ciently close (up to user’s tolerance) to its limit as n goesto infinity can not be determined in advance. An update formula forR!

n from n to n+1 thus appears of much use and this should guide thechoice of (. In order to get such a formula, we study the dependenceon T of

a(T ) =# T

0(( t

T

)A(p(t), q(t))dt.

Di!erentiating with respect to T leads to

da(T )dT

= ((1)A(p(T ), q(T )) " 1T

# T

0

t

T()( t

T

)A(p(t), q(t))dt.(32)

To be of practical use, it is thus necessary that x()(x) is of the form+((x) (where + is an arbitrary constant) so that (32) becomes anordinary di!erential equation for a(T ). The only admissible solutionsare thus monomials in x. We thus consider the following polynomialfilter functions

(p(x) = xp(1 " x)p, p $ N. (33)

Denoting for p and n in N the elementary Riemann sums

Spn =

n&

j=0

( j

n

)pAj ,

it is easy to get the desired update formula

Sp0 = 0 and Sp

n = An + (1 " 1/n)pSpn#1, n 2 1.


100 101 102 10310−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100Stepsize h=0.4

Fig. 1. Error in the averages for p = 1, 3, 5 and h = 0.4 (2D-Kepler problem).

Now, since

(p(x) =p&

k=0

("1)k*

pk

+xp+k and %(p%L1 =

(p!)2

(2p + 1)!

the approximation we seek for can be obtained as the linear combi-nation

R!pn =

(2p + 1)!n(p!)2

p&

k=0

("1)k*

pk

+Sp+k

n .

We now consider the application of our method to the modified2-dimensional Kepler problem with Hamiltonian

H(p, q) = p21 + p2

2 "1'

q21 + q2

2

" µ

('

q21 + q2

2)3.

Besides the Hamiltonian, this system has one other invariant, theangular momentum

L = q2p1 " q1p2.

Our goal is here to estimate the average over the manifold

S = {(p, q) $ R4; L(p, q) = L(p0, q0), H(p, q) = H(p0, q0)}

For µ = 0.2, p0 = (0, 1.1)T and q0 = (1, 0)T this leads to *r+ =1.021466044527120.


10−1 100 101 102 10310−15

10−10

10−5

100Stepsize h=0.05

p=1

p=3

p=5

Fig. 2. Error in the averages for p = 1, 3, 5 and h = 0.05 (2D-Kepler problem).

To this aim, we consider the Verlet method as the basic step anduse the 8th-order 15-stage composition of [13]. In Figure 2 are rep-resented the errors |*r+!p(T ) " *r+| in logarithmic scale for two dif-ferent step-sizes. On the left of the figure, the three curves all reacha plateau corresponding to the hr-error term. Refining the step-sizeremoves this plateau (or at least shifts it to the right, see the rightgraphics). In both cases, the predicted rate of convergence in 1/T p+1

is clearly observed (it corresponds to a slope of p + 1 for (p).

Appendix: some technical results

In this appendix section, we collect a few technical results used in thepaper.

Lemma 3 Let 0 be the function defined on [0, 1] by 0(x) = e#1

x(1!x) .There exist strictly positive constants µ - 1, 1 - (2

&3 + 6)/e2 and

2 - 3 such that the following estimates hold for all k $ N&:

%0(k)%L1 :=

# 1

0|0(k)(x)|dx - µ1kk(k,

%0(k)%L" = sup

x$[0,1]|0(k)(x)| - µ1kk(k.


Proof. Looking for an expression of 0(k)(x) of the form

0(k)(x) =Pk(x)

[8(x)]2ke#

1x(1!x) ,

where 8(x) = x(1 " x) and where Pk is a polynomial, we easily findthe recurrence relation:

P0 0 1 and Pk+1 = 8 ) (1 " 2k8)Pk + P )k 8

2, k 2 0, (34)We now look for bounds on balls Br of radius r > 0 and centerz = 1/2 + 0 i $ C. The bounds for 8 and 8 ) read

supz$Br

|8(z)| - (r2 + 1/4), supz$Br

|8 )(z)| - r,

and the Cauchy integral representation of P )k leads to

/ 9 > 0, supz$Br

|P )k(z)| - r + 9

9sup

z$Br+%

|Pk(z)|.

Inserting these bounds in (34) we get:supz$Br

|Pk+1(z)| - r[k(2r2 " 1/2) + 1] supz$Br

|Pk(z)|

+(r2 + 1/4)2r + 9

9sup

z$Br+%

|Pk(z)|,

-*

r[k(2r2 " 1/2) + 1] +r + 9

9(r2 + 1/4)2

+sup

z$Br+%

|Pk(z)|.

Denoting C(r, k, 9) := r[k(2r2 " 1/2)+ 1]+ r+-- (r2 + 1/4)2, we finally

getsupz$Br

|Pk+1(z)| - C(r, k, 9) supz$Br+%

|Pk(z)|,

- C(r, k, 9)C(r + 9, k " 1, 9) supz$Br+2%

|Pk#1(z)|,

-2

k3

i=0

C(r + i9, k " i, 9)

4

supz$Br+(k+1)%

|P0(z)|.

A bound can then be obtained as follows: let 90 = #1+*

32 , 9 = -0

k andr = 1/2. Then it is easy to check that for all 0 - i - k, we have

C(12

+ i90k

, k " i,90k

) -&

32

[k " i + 1] +1&

3 " 1k + i + 1,

-&

3 + 32

(k + 1),


and hence,2

k3

i=0

C(r + i9, k " i, 9)

4

- [&

3 + 32

(k + 1)]k+1.

Taking into account that P0 0 1, we obtain

/ k $ N&, supz$B1/2

|Pk(z)| - [&

3 + 32

k]k.

It remains to bound 1[.(x)]2k e

# 1x(1!x) . Denoting Y = 1

x(1#x) , we have:

supx$[0,1]

1[8(x)]2k

e#1

x(1!x) = supY (4

e#Y Y 2k

- e#2k(2k)!

-*

4e2

+k

k2k.

Proof of lemma 2. Let us denote by # the operator of backwarddivided di!erences defined by:

/ j $ {0, . . . , n}, #0(j = (j ,

/ j $ {m + 1, . . . , n}, #m+1(j = #m(j "#m(j#1.

The sum in the statement can then be written asn#1&

j=0

(jbj =

n#1&

j=1

bjj&

i=1

#(i +n#1&

j=0

(0bj ,

=1 " bn

1 " b(0 +

n#1&

i=1

#(ibi " bn

1 " b,

=(0 " bn(n#1

1 " b+

11 " b

n#1&

j=1

(#(j)bj = . . . ,

=k&

m=0

bm#m(m " bn#m(n#1

(1 " b)m+1+

1(1 " b)k+1

n#1&

j=k+1

(#k+1(j)bj .

Denoting h = 1/n, it is well-known that, for all n " 1 - j 2 k + 1,there exists :j,k+1 $ [(j " k " 1)h, jh] 5 [0, 1] such that we have:

#k+1(j = ((k+1)(:j,k+1)hk+1


Hence, we can bound the second term in (35) as follows:,,,,,,

n#1&

j=k+1

(#k+1(j)bj

,,,,,,- %((k+1)%

L" hk+1(n " k " 2).

In order to estimate the first sum, we notice that, for 0 - m - k -n " 2,

#m(m = ((m)(:m,m)hm

for some :m,m $ [0,mh] and a Taylor-Lagrange expansion of ((m)(:m,m)at order k + 1 " m gives

#m(m =:km,mhk

(k " m)!((k)(0) +

:k+1m,mhk+1

(k + 1 " m)!((k+1)(5m)

for some 5m $ [0,mh] 5 [0, 1]. Hence, we have:,,,,,

k&

m=0

bm

(1 " b)m#m(m

,,,,, - |((k)(0)| kkhk

|1 " b|k+1

k&

m=0

|1 " b|m

(m)!

+%((k+1)%L"

kkhk+1

|1 " b|

k&

m=0

|1 " b|m(m + 1)!

,

- e2kkhk

|1 " b|k+1

(|((k)(0)| + h%((k+1)%

L"

).

Similarly we have:#m(n#1 = ((m)(:n#1,m)hm

for some :n#1,m $ [1 " (m + 1)h, 1 " h] 5 [0, 1], so that,,,,,

k&

m=0

bn

(1 " b)m#m(n#1

,,,,, -2e2kkhk

|1 " b|k+1

(|((k)(1)| + h%((k+1)%L"

).

Gathering the contributions of all terms then gives the result.

Acknowledgements The authors are glad to thank Christian Lubich for stimu-lating discussions on the subject of this paper, particularly for suggesting the useof general filtered averages rather than just iterated averages. We also gratefullyacknowledge the financial support of INRIA through the contract grant “Actionde Recherche Concertee” PRESTISSIMO.

References

1. V.I. Arnold. Small denominators and problems of stability of motion in clas-sical and celestial mechanics, Russian Math. Surveys 18 (1963) 85-191.


2. V.I. Arnold. Mathematical methods of classical mechanics, volume 60 of Grad-uate Texts in Mathematics. Springer-Verlag, Berlin, 1978.

3. E. Cances, F. Castella, P. Chartier, E. Faou, C. Le Bris, F. Legoll and G.Turinici, High-order averaging schemes with error bounds for thermodynami-cal properties calculations by MD simulations, submitted to J. Chem. Phys.,2003.

4. M.P. Do Carmo. Riemannian Geometry, Series Mathematics: Theory andApplications. Birkhauser, Boston, 1992.

5. E. Hairer, C. Lubich, and G. Wanner. Geometric numerical integration, vol-ume 31 of Springer Series in Computational Mathematics. Springer-Verlag,Berlin, 2002. Structure-preserving algorithms for ordinary di!erential equa-tions.

6. A.N. Kolmogorov. On conservation of conditionally periodic motions undersmall perturbations of the Hamiltonian, Dokl. Akad. Nauk SSSR 98 (1954)527-530

7. A.N. Kolmogorov. General theory of dynamical systems and classical mechan-ics, Proc. Int. Congr. Math. Amsterdam 1954, Vol. 1, 315-333.

8. J. Moser. On invariant curves of area-preserving mappings of an annulus,Nachr. Akad. Wiss. Gottingen, II. Math.-Phys. K1. 1962, 1-20.

9. N.N. Nekhoroshev, An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems, Russ. Math. Surveys 32 (1977), 1-65.

10. A. Papoulis. Signal Analysis, Electrical and Electronic Engineering Series.McGraw-Hill, Singapore, 1984.

11. Z. Shang. KAM theorem of symplectic algorithms for Hamiltonian systems,Numer. Math. 83 (1999) 477-496.

12. Z. Shang. Resonant and Diophantine step sizes in computing invariant toriof Hamiltonian systems, Nonlinearity 13 (2000) 299-308.

13. M. Suzuki and K. Umeno. Higher-order decomposition theory of exponentialoperators and its applications to QMC and nonlinear dynamics. In: ComputerSimulation Studies in Condensed-Matter Physics VI, Landau, Mon, Schuttler(eds.), Springer Proceedings in Physics 76 (1993) 74-86.

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