gni.ps

GniCodes { Matlab Programs for

Geometri Numeri al Integration

Ernst Hairer

1

and Martin Hairer

2

1

Se tion de math�ematiques, Univ. Gen�eve, CH-1211 Gen�eve 24, Switzerland

2

Mathemati s Institute, Univ. Warwi k, Coventry CV4 7AL, England

Abstra t. Geometri numeri al integration is synonymous with stru ture-preser-

ving integration of ordinary di�erential equations. These notes, prepared for the

Durham summer s hool 2002, are omplementary to the monograph of Hairer,

Lubi h and Wanner [12℄. They give an introdu tion to the subje t, and they dis uss

and explain the use of Matlab programs for experimenting with stru ture-preserving

algorithms.

We start with presenting some typi al lasses of problems having properties

that are important to be onserved by the dis retization (Se tion 1). The ow of

Hamiltonian di�erential equations is symple ti and possesses onserved quanti-

ties. Conservative systems have a time-reversible ow. Di�erential equations with

�rst integrals and problems on manifolds are also onsidered. We then introdu e

in Se tion 2 simple symple ti and symmetri integrators, (partitioned) Runge-

Kutta methods, omposition and splitting methods, linear multistep methods, and

algorithms for Hamiltonian problems on manifolds. We brie y dis uss their sym-

ple ti ity and symmetry. The improved performan e of su h geometri integrators

is best understood with the help of a ba kward error analysis (Se tion 3). We ex-

plain some impli ations for the long-time integration of Hamiltonian systems and

of ompletely integrable problems.

Se tion 4 is devoted to a presentation and explanation of Matlab odes for

impli it Runge-Kutta, omposition, and multistep methods. The �nal Se tion 5

gives a omparison of the di�erent methods and illustrates the use of these programs

at some typi al interesting situations: the omputation of Poin ar�e se tions, and

the simulation of the motion of two bodies on a sphere. The Matlab odes as well

as their Fortran 77 ounterparts an be downloaded at

http://www.unige. h/math/folks/hairer

under the item \software".

1 Problems to be Solved

For the numeri al solution of ordinary di�erential equations there exist well-

developed theories, and ex ellent general purpose odes are available and

widely used. If the ow of the di�erential equation has a parti ular stru -

ture, then its preservation by the dis retization s heme an improve on-

siderably its performan e and its qualitative behaviour. This arti le fo uses

on stru ture-preserving algorithms for some important lasses of problems {

Hamiltonian systems and reversible di�erential equations.

2 Ernst Hairer and Martin Hairer

1.1 Hamiltonian Systems

For a smooth fun tion H(p; q) de�ned on an open set D � R

d

� R

d

we

onsider the di�erential equation

_p

i

= �

�H

�q

i

�

p; q

�

; _q

i

=

�H

�p

i

�

p; q

�

; i = 1; : : : ; d: (1.1)

The dimension d of the ve tors p and q is alled the `degree of freedom' of

the system. We also use the more ompa t notation

_p = �r

q

H(p; q); _q = r

p

H(p; q);

or

_y = J

�1

rH(y); y =

�

p

q

�

; J =

�

0 I

�I 0

�

; (1.2)

where r

p

H;r

q

H;rH denote the olumn ve tors of partial derivatives with

respe t to the omponents of p; q; y, respe tively. The matrix J is the stru -

ture matrix of Hamiltonian systems in anoni al form.

Throughout this arti le we denote by '

t

(y) the exa t ow of the system

(1.2), i.e., '

t

(y

0

) = y(t) is the solution at time t of the problem (1.2) with

initial value y(0) = y

0

.

Example 1.1 (Classi al Me hani al Systems). Consider a me hani al system

that an be des ribed with (minimal) oordinates q 2 R

d

. Denote its kineti

energy by T (q; _q) =

1

2

_q

T

M(q) _q (with a symmetri positive de�nite matrix

M(q)) and its potential energy by U(q). The motion of the system is then

given as the solution of the Euler-Lagrange equations

d

dt

�

M(q) _q

�

=

�

�q

�

T (q; _q)� U(q)

�

(1.3)

orresponding to the variational problem

R �

T (q; _q) � U(q)

�

dt ! min. In-

trodu ing the new variables p :=M(q) _q (momenta or Poisson variables) the

di�erential equation (1.3) is equivalent to the Hamiltonian system (1.1) with

H(p; q) =

1

2

p

T

M(q)

�1

p+ U(q): (1.4)

This is an immediate onsequen e from omputing the partial derivatives of

this fun tion H(p; q). A simple example, often used for illustrations, is the

mathemati al pendulum for whi h the Hamiltonian is

H(p; q) =

1

2

p

2

� os q: (1.5)

Due to their spe ial stru ture, Hamiltonian systems have several interest-

ing properties:

GniCodes { Matlab Programs for Geometri Numeri al Integration 3

−3 −2 −1 0 1 2 3 4

−2

−1

1

2

−3 −2 −1 0 1 2 3 4

−2

−1

1

2

A

'

�=3

(A)

Fig. 1.1. Level urves H(p; q) = onst for the pendulum problem (left pi ture),

and area-preservation of its exa t ow (right pi ture).

� the Hamiltonian H(p; q) is onstant along solutions of (1.1); for lassi al

me hani al systems this means that the total energy (sum of kineti and

potential energies) is a onserved quantity.

� for systems with one degree of freedom the ow '

t

is area-preserving; for

the general ase it is volume-preserving. This means that

�

�

'

t

(A)

�

= �(A) for t � 0 (1.6)

for any ompa t set A � R

d

�R

d

.

� the ow '

t

is a symple ti transformation, i.e.,

'

0

t

(y)

T

J '

0

t

(y) = J for t � 0; (1.7)

where the prime in '

0

t

(y) denotes the derivation with respe t to y.

The �rst property is immediately veri�ed by di�erentiating

d

dt

H

�

p(t); q(t)

�

=

: : : = 0. The solutions of the mathemati al pendulum are therefore on the

level urves of the Hamiltonian (1.5); see Fig. 1.1. The se ond property is

a onsequen e of the third, be ause (1.7) and the ontinuous dependen e of

'

0

t

(y) on t imply det'

0

t

(y) = 1. This together with the transformation formula

for multiple integrals proves (1.6). The right pi ture of Fig. 1.1 illustrates the

area-preservation of the exa t ow for the pendulum equation.

The symple ti ity ondition (1.7) has a ni e geometri interpretation. It

is equivalent to the property that

!

�

'

t

(A)

�

= !(A) for t � 0

holds for any two-dimensional sub-manifoldA ofR

d

�R

d

, where !(A) denotes

the sum of the oriented areas of the proje tions of A onto the (p

i

; q

i

)-plane.

The important feature is that this property is hara teristi for Hamiltonian

systems ( f. [12, hap.VI℄), whi h means that whenever the ow of a di�er-

ential equation _y = f(y) is symple ti for all t and all y, then f(y) is lo ally

of the form f(y) = J

�1

rH(y). This hara teristi property of Hamiltonian

systems motivates the sear h for dis retizations that are symple ti .


1.2 Reversible Di�erential Equations

Consider �rst a me hani al system for whi h the equations of motion are given

by the se ond order di�erential equation (1.3). Sin e T (q; _q) is quadrati in

_q, they are equivalent to the system

_q = v; _v = g(q; v); (1.8)

satisfying g(q;�v) = g(q; v). This implies the time-reversibility of the system

and means that whenever

�

q(t); v(t)

�

is a solution of (1.8), also

�

q(�t);�v(�t)

�

is a solution. For example, in the study of planetary motion, the same dif-

ferential equation permits us to investigate the future and the past, one only

has to hange the sign of the velo ity ve tor v.

More generally, we onsider a di�erential equation _y = f(y) and a linear

invertible transformation �. We all the di�erential equation �-reversible if

(� Æ f)(y) = �(f Æ �)(y): (1.9)

For the previous situation we have y = (q; v), �(q; v) = (q;�v), and the ve tor

�eld f(y) =

�

v; g(q; v)

�

indeed satis�es (1.9) whenever g(q;�v) = g(q; v).

This is illustrated in the left pi ture of Fig. 1.2 at the hand of the perturbed

pendulum equation _q = v; _v = � sin q � v

2

=5, whi h is still �-reversible with

respe t to �(q; v) = (q;�v), but whi h is no longer Hamiltonian.

The ow of a �-reversible di�erential equation has a remarkable property:

� it is �-reversible, i.e., it satis�es (see the right pi ture of Fig. 1.2)

(� Æ '

t

)(y) = ('

�1

t

Æ �)(y) for all t and all y. (1.10)

The proof of this statement is straightforward. One he ks by di�erentiation

that (�Æ'

t

)(y) and ('

�1

t

Æ�)(y) = ('

�t

Æ�)(y) are both solutions of the same

di�erential equation _z = �f(z), and are identi al for t = 0. Formula (1.10)

−2 −1 0 1 2

−1

1

� �

f(y)

�f(y)

f(�y)

−2 −1 0 1 2

−1

1

� �

'

t

'

�1

t

Fig. 1.2. The �-reversibility of the ve tor �eld f(q; v) = (v;� sin q � v

2

=5)

T

(left

pi ture), and the �-reversibility of the orresponding ow.


thus follows from the uniqueness of the solution of an initial value problem.

Analogous to the situation of Hamiltonian problems, this property is hara -

teristi for �-reversible di�erential equations. This means that whenever the

ow of a di�erential equation _y = f(y) satis�es (1.10), then (1.9) holds. It is

thus natural to look for numeri al methods that share this property.

Example 1.2 (Kepler Problem). The relative motion of two bodies whi h at-

tra t ea h other is des ribed by the di�erential equation

_q

1

= v

1

; _q

2

= v

2

; _v

1

= �

q

1

(q

2

1

+ q

2

2

)

3=2

; _v

2

= �

q

2

(q

2

1

+ q

2

2

)

3=2

: (1.11)

Sin e it an be onsidered as a lassi al me hani al system, it is �-reversible

for �(q

1

; q

2

; v

1

; v

2

) = (q

1

; q

2

;�v

1

;�v

2

). However, there are more symmetries

in this problem, and it is seen to be �-reversible also for �(q

1

; q

2

; v

1

; v

2

) =

(q

1

;�q

2

;�v

1

; v

2

).

Example 1.3 (Se ond Order Di�erential Equations). Many problems of pra -

ti al appli ations lead to �q = g(q), or equivalently,

_q = v; _v = g(q): (1.12)

For example, all lassi al me hani al systems for whi h M(q) =M is a on-

stant matrix are of this form. The di�erential equation (1.12) is �-reversible

for �(q; v) = (q;�v) independent of the form of g(q). It is Hamiltonian only

if g(q) = �r

q

U(q) for some potential fun tion U(q).

1.3 Hamiltonian and Reversible Systems on Manifolds

It is often diÆ ult to �nd suitable minimal oordinates for des ribing the

motion of me hani al systems. Moreover, minimal oordinates are in general

de�ned only lo ally and frequent hanges of harts may be ne essary. To avoid

this diÆ ulty we onsider oordinates q 2 R

d

that are subje t to onstraints

g(q) = 0. Expressing the Euler-Lagrange equations and their Hamiltonian

formulation in terms of these oordinates, we are led to a system of the form

_p = �r

q

H(p; q)�r

q

g(q)�

_q = r

p

H(p; q); 0 = g(q);

(1.13)

where the additional term with the Lagrange multiplier � for es the solution

to satisfy g(q) = 0. Here, p and q are ve tors inR

d

, g(q) = (g

1

(q); : : : ; g

m

(q))

T

is the ve tor of onstraints, and r

q

g =

�

r

q

g

1

; : : : ;r

q

g

m

�

is the transposed

Ja obian matrix of g(q).

Di�erentiating the onstraint 0 = g

�

q(t)

�

with respe t to time yields

0 = r

q

g(q)

T

r

p

H(p; q) (1.14)


(the so- alled hidden onstraint) whi h is an invariant of the ow of (1.13).

A se ond di�erentiation gives the relation

0 =

�

�q

�

r

q

g(q)

T

r

p

H(p; q)

�

r

p

H(p; q)

�r

q

g(q)

T

r

2

p

H(p; q)

�

r

q

H(p; q) +r

q

g(q)�

�

;

(1.15)

whi h allows us to express � in terms of (p; q), if the matrix

r

q

g(q)

T

r

2

p

H(p; q)r

q

g(q) is invertible (1.16)

(r

2

p

H denotes the Hessian matrix of H). Inserting the so-obtained fun tion

�(p; q) into (1.13) gives a di�erential equation for (p; q) on the manifold

M =

�

(p; q) j g(q) = 0; r

q

g(q)

T

r

p

H(p; q) = 0

: (1.17)

This interpretation allow us to dedu e the existen e and uniqueness of the

solution from the standard theory for ordinary di�erential equations, provided

that the initial values satisfy (p

0

; q

0

) 2M.

Important properties of the system (1.13) that should be onserved by a

dis retization are the following:

� for (p

0

; q

0

) 2 M the solution stays on the manifold M for all t; hen e,

the ow is a mapping '

t

:M!M.

� the ow '

t

is a symple ti transformation on M whi h means that

('

0

t

(y)�)

T

J '

0

t

(y)� = �

T

J � for �; � 2 T

y

M; (1.18)

here, the produ t '

0

t

(y)� has to be interpreted as the dire tional deriva-

tive.

� for Hamiltonians satisfying H(�p; q) = H(p; q) the ow '

t

is �-reversible

for �(p; q) = (�p; q) in the sense that (1.10) holds for y = (p; q) 2 M.

Example 1.4 (Two-Body Problem on the Sphere). We are interested in the

motion of two bodies whi h attra t ea h other, but whi h are restri ted to

stay on a sphere. Using Cartesian oordinates q

1

; q

2

2 R

3

for the positions of

the two bodies and p

1

; p

2

2 R

3

for their velo ities, the Hamiltonian be omes

(after a suitable normalization)

H(p

1

; p

2

; q

1

; q

2

) =

1

2

�

p

T

1

p

1

+ p

T

2

p

2

�

+ U(q

1

; q

2

); (1.19)

and the onstraint equations g(q

1

; q

2

) = 0 with g : R

6

! R

2

are given by

q

T

1

q

1

� 1 = 0; q

T

2

q

2

� 1 = 0: (1.20)


Fig. 1.3. A solution of the two-body problem on the sphere; initial values are

indi ated by larger symbols; the geodesi onne tion between the two bodies is

plotted at every se ond time step in the right pi ture.

A ording to Kozlov and Harin [18℄, we hoose U(q

1

; q

2

) = � os#= sin# as

potential, where # is the distan e between the two bodies along a geodesi s.

We have os# = q

T

1

q

2

, so that the equations of motion be ome

_q

1

= p

1

; _p

1

= f(q

T

1

q

2

)q

2

� �

1

q

1

;

_q

2

= p

2

; _p

2

= f(q

T

1

q

2

)q

1

� �

2

q

2

;

(1.21)

together with (1.20), where

f( ) =

1

(1�

2

)

3=2

:

The initial values have to lie on the manifold

M =

�

(p

1

; p

2

; q

1

; q

2

) ; q

T

1

q

1

= 1; q

T

2

q

2

= 1; q

T

1

p

1

= 0; q

T

2

p

2

= 0

;

and the solution stays on M for all t.

A parti ular solution is plotted in Fig. 1.3. We have hosen

q

i

=

�

os�

i

sin �

i

; sin�

i

sin �

i

; os �

i

�

T

with (�

1

; �

1

) = (0:8; 0:6) and (�

2

; �

2

) = (0:5; 1:5) as initial values for the

positions, and

p

i

=

�

�

_

�

i

sin�

i

sin �

i

+

_

�

i

os�

i

os �

i

;

_

�

i

os�

i

sin �

i

+

_

�

i

sin�

i

os �

i

;�

_

�

i

sin �

i

)

with (

_

�

1

;

_

�

1

) = (1:1;�0:2) and (

_

�

2

;

_

�

2

) = (�0:8; 0:0) as initial values for the

velo ities. The two bodies are indi ated by small squares in di�erent olors.

The right pi ture of Fig. 1.3 also shows the geodesi onne tion between the

two bodies.


Example 1.5 (Rigid Body Simulation). The motion of a rigid body with a

�xed point hosen at the origin an be des ribed by an orthogonal matrix

Q(t). Denoting by I

1

; I

2

; I

3

the moments of inertia of the body, its kineti

energy is

T =

1

2

�

I

1

2

1

+ I

2

2

2

+ I

3

2

3

�

;

where the angular velo ity = (

1

;

2

;

3

)

T

of the body is de�ned by

b

=

0

�

0 �

3

2

3

0 �

1

�

2

1

0

1

A

= Q

T

_

Q;

(see [2, Chap. 6℄). In terms ofQ, the kineti energy on the manifold fQ jQ

T

Q =

Ig be omes

T =

1

2

tra e (

b

D

b

T

) =

1

2

tra e (Q

T

_

QD

_

Q

T

Q) =

1

2

tra e (

_

QD

_

Q

T

);

where D = diag (d

1

; d

2

; d

3

) is given by the relations I

1

= d

2

+d

3

, I

2

= d

3

+d

1

,

and I

3

= d

1

+ d

2

. With P = �T=�

_

Q =

_

QD, we are thus on erned with

H(P;Q) =

1

2

tra e (PD

�1

P

T

) + U(Q);

and the onstrained Hamiltonian system be omes

_

P = �r

Q

U(Q)�Q�;

_

Q = PD

�1

; 0 = Q

T

Q� I;

(1.22)

where � is a symmetri matrix onsisting of Lagrange multipliers. This is of

the form (1.13) and satis�es the regularity ondition (1.16).

2 Symple ti and Symmetri Integrators

A numeri al integrator is a family �

h

(y) of maps on the phase spa e that

approximates the exa t ow '

h

(y) of the di�erential equation. It is the aim

of `geometri integration' to onstru t and to study methods for whi h the

numeri al solution, given by y

n+1

= �

h

(y

n

), preserves the stru ture of the

problem. We are mainly interested in methods for whi h �

h

is symple ti or

�-reversible, when it is applied to a Hamiltonian or �-reversible di�erential

equation, respe tively.

2.1 Simple Symple ti Methods

The simplest numeri al methods for general di�erential equations _y = f(y)

are the expli it Euler method

y

n+1

= y

n

+ hf(y

n

) (2.1)


−2 0 2 4

−2

2

−2 0 2 4

−2

2

−2 0 2 4

−2

2

−2 0 2 4

−2

2

−2 0 2 4

−2

2

−2 0 2 4

−2

2

−2 0 2 4

−2

2

−2 0 2 4

−2

2

expli it Euler impli it Euler

symple ti Euler (2.3) symple ti Euler (2.4)

Fig. 2.1. Numeri al ow with step size h = �=3 for the four Èuler methods' of

Se t. 2.1. The exa t ow is in luded as a white shadow.

and the impli it Euler method

y

n+1

= y

n

+ hf(y

n+1

): (2.2)

Here, h is the step size, and y

n

is an approximation to the solution y(t) at

time t = nh. For Hamiltonian systems (1.1) we onsider the method

p

n+1

= p

n

� hr

q

H(p

n+1

; q

n

); q

n+1

= q

n

+ hr

p

H(p

n+1

; q

n

); (2.3)

whi h treats the p-variable by the impli it Euler method and the q-variable

by the expli it Euler method. Similarly, we also onsider

p

n+1

= p

n

� hr

q

H(p

n

; q

n+1

); q

n+1

= q

n

+ hr

p

H(p

n

; q

n+1

): (2.4)

Both methods are alled symple ti Euler method.

Example 2.1. We apply all four methods to the pendulum problem whi h is

Hamiltonian with H(p; q) given by (1.5), and we onsider initial values in the

set A of Fig. 1.1. The numeri al solution obtained with the large step size

h = �=3 is illustrated in Fig. 2.1. Neither the expli it nor the impli it Euler

methods are area-preserving (i.e., symple ti ). We shall see in the following

theorem that both `symple ti Euler methods' are area-preserving (hen e

the name symple ti ). Due to the large step size, however, the numeri al

solution di�ers signi� antly from the exa t solution whi h is in luded as a

white shadow in the pi tures ( ompare with Fig. 1.1).


Theorem 2.1. For the numeri al s hemes (2.3) and (2.4) the mapping

�

h

:

�

p

n

q

n

�

7!

�

p

n+1

q

n+1

�

is a symple ti transformation.

The proof of this theorem is straightforward (de Vogelaere [42℄ and [12,

p. 176℄). One omputes the Ja obian of �

h

by impli it di�erentiation, and

one he ks the identity (1.7).

2.2 Simple Reversible Methods

We next onsider �-reversible di�erential equations (i.e., � Æ f = �f Æ �) as

dis ussed in Se t. 1.2.

Theorem 2.2. If a numeri al method �

h

satis�es

� Æ �

h

= �

�h

Æ � and �

h

= �

�1

�h

; (2.5)

then it is �-reversible, i.e., � Æ �

h

= �

�1

h

Æ �.

This statement is obvious. The interest of this theorem lies in the fa t that

the se ond ondition of (2.5) is independent of �, whereas the �rst ondition of

(2.5) is easy to he k and satis�ed by all `reasonable' methods. For example,

the expli it Euler dis retization (2.1) yields

(� Æ �

h

)(y

n

) = �y

n+1

= �y

n

+ h�f(y

n

) = �y

n

� hf(�y

n

) = (�

�h

Æ �)(y

n

);

and a similar simple omputation shows that the impli it Euler method and

all (expli it and impli it) Runge-Kutta methods satisfy the �rst ondition of

(2.5). For partitioned Runge-Kutta methods, su h as the symple ti Euler

s heme, this is true for transformations � whi h are of the form �(q; v) =

(�

1

(q); �

2

(v)).

If �

h

(y) represents a numeri al method of order at least one, i.e., �

h

(y) =

y + hf(y) +O(h

2

), then also �

�1

�h

(y) = y + hf(y) +O(h

2

) and

�

�

h

:= �

�1

�h

(2.6)

is a numeri al method of order at least one. It is alled the adjoint method of

�

h

. Whenever an integrator satis�es

�

�

h

= �

h

; (2.7)

it is alled a symmetri method. The se ond ondition in (2.5) of Theorem 2.2

is thus equivalent to the symmetry of the method �

h

.

Ex hanging h$ �h and y

n

$ y

n+1

in (2.1) shows that the adjoint of the

expli it Euler method is the impli it Euler method and vi e versa. Similarly,


the adjoint of the symple ti Euler method (2.3) is the method (2.4). None

of these methods is symmetri .

Using the notion of the adjoint method it is easy to onstru t symmet-

ri methods: let

h

be an arbitrary method of order at least one, then the

ompositions

h=2

Æ

�

h=2

and

�

h=2

Æ

h=2

(2.8)

are symmetri methods of order at least two. The symmetry follows from the

properties (�

h

Æ

h

)

�

=

�

h

Æ �

�

h

and (�

�

h

)

�

= �

h

, and order at least two is a

onsequen e of the fa t that symmetri method always have an even order.

For example, if we let

h

be the expli it Euler method, then the methods

of (2.8) are

y

n+1

= y

n

+ h f

�

y

n

+ y

n+1

2

�

; (2.9)

the impli it midpoint rule, and

y

n+1

= y

n

+

h

2

�

f(y

n

) + f(y

n+1

)

�

; (2.10)

the trapezoidal rule, respe tively.

2.3 St�ormer/Verlet S heme

We next onsider Hamiltonian systems (1.1) and the symple ti Euler method

(2.3) in the role of

h

. The ompositions (2.8) then yield

q

n+1=2

= q

n

+

h

2

r

p

H(p

n

; q

n+1=2

)

p

n+1

= p

n

�

h

2

�

r

q

H(p

n

; q

n+1=2

) +r

q

H(p

n+1

; q

n+1=2

)

�

q

n+1

= q

n+1=2

+

h

2

r

p

H(p

n+1

; q

n+1=2

)

(2.11)

and

p

n+1=2

= p

n

�

h

2

r

q

H(p

n+1=2

; q

n

)

q

n+1

= q

n

+

h

2

�

r

p

H(p

n+1=2

; q

n

) +r

p

H(p

n+1=2

; q

n+1

)

�

p

n+1

= p

n+1=2

�

h

2

r

q

H(p

n+1=2

; q

n+1

)

(2.12)

respe tively. For the important spe ial ase H(p; q) =

1

2

p

2

+ U(q), method

(2.12) redu es to (after elimination of the p-variable)

q

n+1

� 2q

n

+ q

n�1

= �h

2

r

q

U(q

n

): (2.13)

This dis retization of �q = �r

q

U(q) is attributed to Newton ( f. [13℄), De-

lambre ( f. [25℄), En ke, St�ormer [36℄, and Verlet [41℄. The methods (2.11)

and (2.12) are nowadays often alled St�ormer/Verlet s heme.

Let us olle t the most important properties of the St�ormer/Verlet s heme:


−2 0 2 4

−2

2

−2 0 2 4

−2

2

−2 0 2 4

−2

2

−2 0 2 4

−2

2

St�ormer/Verlet (2.11) St�ormer/Verlet (2.12)

Fig. 2.2. Numeri al ow with step size h = �=3 for the two versions of the

St�ormer/Verlet method. The exa t ow is in luded as a white shadow.

� the method is of order two,

� it is a symple ti method,

� it is a symmetri method,

� for separable Hamiltonians T (p) + U(q) the method is expli it,

� the method exa tly onserves quadrati �rst integrals p

T

Cq, e.g., the

angular momentum in N -body problems.

The �rst four statements are immediate onsequen es of the above dis us-

sions. A proof of the last property is given in [12, p. 98℄.

In Fig. 2.2 we repeat the experiment of Example 2.1, but this time with

the two versions of the St�ormer/Verlet method. We learly observe the higher

a ura y ( ompared to the �rst order methods) and the area-preservation.

The St�ormer/Verlet s heme is an ex ellent geometri integrator and it

is widely used, in parti ular in mole ular dynami s where a orre t qual-

itative simulation is of utmost importan e. For long-time omputations in

astronomy, where a very high a ura y is demanded, the order two of the

St�ormer/Verlet s heme is too low.

2.4 Splitting Methods

A di�erent approa h for onstru ting simple geometri integrators is based

on the idea of splitting the ve tor �eld as

_y = f

[1℄

(y) + f

[2℄

(y): (2.14)

If by han e the exa t ows '

[1℄

t

and '

[2℄

t

of the systems _y = f

[1℄

(y) and

_y = f

[2℄

(y) an be al ulated exa tly, we an onsider

�

h

= '

[1℄

h

Æ '

[2℄

h

(2.15)

as simple numeri al integrator. It follows from Taylor expansion that this

method is of order one. Even more important is the symmetri (se ond order)

omposition

�

h

= '

[1℄

h=2

Æ '

[2℄

h

Æ '

[1℄

h=2

(2.16)


whi h is usually alled Strang splitting . These splitting methods have the

following obvious properties:

� if both, f

[1℄

(y) and f

[2℄

(y), are Hamiltonian ve tor �elds, then the om-

positions (2.15) and (2.16) are symple ti integrators;

� if both, f

[1℄

(y) and f

[2℄

(y), are �-reversible, then the symmetri method

(2.16) is �-reversible.

For some situations the splitting (2.14) is obvious. For example, if a Hamil-

tonian system has H(p; q) = T (p) + U(q) as Hamiltonian, then the ows

orresponding to H

[1℄

(p; q) = T (p) and H

[2℄

(p; q) = U(q) are given expli itly

by

'

[1℄

t

(p; q) =

�

p; q + tr

p

T (p)

�

; '

[2℄

t

(p; q) =

�

p� tr

q

U(q); q

�

:

The resulting splitting methods (2.15) and (2.16) are then equivalent to the

symple ti Euler method (2.3) and to the St�ormer/Verlet s heme (2.12), re-

spe tively. In general, however, it is an art to �nd a suitable splitting ( f. [25℄).

2.5 High Order Geometri Integrators

We start this se tion with a numeri al experiment that motivates the sear h

for high order symple ti and symmetri numeri al integrators. We onsider

the Kepler problem whi h is Hamiltonian with

H(p

1

; p

2

; q

1

; q

2

) =

1

2

(p

2

1

+ p

2

2

)�

1

p

q

2

1

+ q

2

2

; (2.17)

and we take as initial values

q

1

(0) = 1� e; q

2

(0) = 0; p

1

(0) = 0; p

2

(0) =

p

(1 + e)(1� e)

�1

;

su h that the solution is an ellipse with e entri ity e = 0:6. Figure 2.3 shows

the work pre ision diagrams (global error at the endpoint after 200 revolu-

tions against the required number of fun tion evaluations and the omputer

time, respe tively) for the se ond order St�ormer/Verlet s heme as well as for

various methods of order eight. It learly demonstrates that for high a u-

ra y requirements (say 10 digits) the low order method annot ompete with

the high order ones. It would need about 1000 times more pu time. The

irregularities at the right bottom orner of the pi tures are due to round-o�.

Impli it Runge-Kutta Methods In the end of the 19th and the beginning

of the 20th entury Runge [30℄ and Kutta [19℄ introdu ed generalizations of

the expli it Euler method with the aim of getting more a urate numeri al

approximations. These expli it methods an neither be symple ti nor sym-

metri as follows from the hara terizations given below. Mu h more impor-

tant for our purpose are impli it Runge-Kutta methods, introdu ed mainly


105 10610−12

10−9

10−6

10−3

100

10−1 10010−12

10−9

10−6

10−3

100

e

r

r

o

r

f n. eval.

omp

irk2

lmm2

verlet

e

r

r

o

r

pu time

irk2

omp

lmm2

verlet

Fig. 2.3. Work pre ision diagrams for the St�ormer/Verlet s heme and for three

methods of order eight; impli it Runge-Kutta method (irk2), omposition method

( omp), and linear multistep method (lmm2).

in the work of But her [5℄. For Hamiltonian systems or for general partitioned

di�erential equations

_q = f(q; v); _v = g(q; v) (2.18)

we onsider so- alled partitioned Runge-Kutta methods, whi h treat the om-

ponents of q and those of v by possibly di�erent impli it Runge-Kutta meth-

ods. They are de�ned by

k

i

= f

�

q

n

+ h

s

X

j=1

a

ij

k

j

; v

n

+ h

s

X

j=1

ba

ij

`

j

�

;

`

i

= g

�

q

n

+ h

s

X

j=1

a

ij

k

j

; v

n

+ h

s

X

j=1

ba

ij

`

j

�

;

q

n+1

= q

n

+ h

s

X

i=1

b

i

k

i

; v

n+1

= v

n

+ h

s

X

i=1

b

b

i

`

i

:

(2.19)

The equations for k

i

; `

i

(i = 1; : : : ; s) are nonlinear and have to be solved by

�xed-point iteration, provided that the step size h is suÆ iently small.

It turns out that the method (2.19) is symple ti for general Hamiltonian

systems, if the following relations are satis�ed:

b

i

ba

ij

+

b

b

j

a

ji

= b

i

b

b

j

for i; j = 1; : : : ; s;

b

i

=

b

b

i

for i = 1; : : : ; s:

(2.20)

It is symmetri , if

a

s+1�i;s+1�j

+ a

ij

= b

j

for all i; j,

ba

s+1�i;s+1�j

+ ba

ij

=

b

b

j

for all i; j.

(2.21)


If the method does not ontain super uous stages and if the stages are suit-

ably ordered, the onditions (2.20) and (2.21) are also ne essary for symple -

ti ity and symmetry, respe tively. These hara terizations have been obtained

originally by Lasagni [22℄, Sanz-Serna [32℄ and Suris [37℄ for symple ti ity,

and by Stetter [35℄ and Wanner [43℄ for symmetry. They are dis ussed in

detail in Chapters V and VI of [12℄.

For the important spe ial ase �q = g(q), i.e., f(q; v) = v in (2.18) and

g(q; v) independent of v, the variables k

i

an be eliminated expli itly and the

method (2.19) redu es to

`

i

= g

�

q

n

+ h

i

v

n

+ h

2

s

X

j=1

ea

ij

`

j

�

; i = 1; : : : s;

q

n+1

= q

n

+ hv

n

+ h

2

s

X

i=1

e

b

i

`

i

; v

n+1

= v

n

+ h

s

X

i=1

b

b

i

`

i

:

(2.22)

where

i

=

P

s

j=1

a

ij

, and

e

b

i

, ea

ij

are the oeÆ ients of

e

b

T

= b

T

b

A and

e

A = A

b

A.

Example 2.2 (Method Used as ìrk2' in Fig. 2.3). The most important sym-

ple ti impli it Runge-Kutta methods are the so- alled Gauss methods. They

are built on the Gaussian quadrature (b

i

;

i

)

s

i=1

, whi h is interpolatory and

for whi h

1

; : : : ;

s

are the zeros of the sth shifted Legendre polynomial

d

s

dx

s

�

x

s

(x� 1)

s

�

:

The oeÆ ients a

ij

are omputed from the linear system

s

X

j=1

a

ij

k�1

j

=

k

i

k

for i; k = 1; : : : ; s.

We let

b

b

i

= b

i

and ba

ij

= a

ij

in (2.19), so that all omponents of the di�erential

equation are treated by the same method.

The method obtained in this way has originally been introdu ed by But her

[6℄, and it has many ni e properties. It is of order 2s (whi h is maximal among

all s-stage Runge-Kutta methods), it is symple ti and symmetri , so that

it is extremely well suited in the ontext of geometri integration. The only

disadvantage is that even for simple situations su h as �q = g(q), it gives an

impli it dis retization. In the experiment of Fig. 2.3 we use this method with

s = 4. The `2' in ìrk2' indi ates that the ode is for se ond order di�erential

equations �q = g(q) only, and that it is implemented as (2.22).

Partitioned Multistep Methods Another extension of the Euler methods

are linear multistep methods, originally introdu ed by Adams in 1855 and

published in Bashforth [3℄. Neither expli it nor impli it lassi al multistep


methods have been su essful in geometri integration. Lambert and Watson

[21℄ onsidered spe ial lasses for se ond order di�erential equations �q =

g(q), whi h have been revived by Quinlan and Tremaine [27℄ for the long-

time integration of planetary orbits. For partitioned di�erential equations

(2.18), whi h are more general than _q = v; _v = g(q), these methods an be

interpreted as partitioned linear multistep methods, de�ned by

k

X

j=0

�

j

q

n+j

= h

k

X

j=0

�

j

f(q

n+j

; v

n+j

);

b

k

X

j=0

b�

j

v

n+j

= h

b

k

X

j=0

b

�

j

g(q

n+j

; v

n+j

):

(2.23)

It is not evident to dis uss symple ti ity and symmetry of multistep methods,

be ause we are on erned with an algorithm (y

n

; : : : ; y

n+k�1

) 7! y

n+k

and

not with a one-step method y

n+1

= �

h

(y

n

) whi h is a transformation on

the phase spa e. However, Kir hgraber [17℄ showed that to every onsistent

stri tly stable multistep method one an asso iate a so- alled underlying one-

step method �

h

whi h has the same long-time dynami s. More pre isely, it

satis�es the following properties:

� for every y

0

, the sequen e de�ned by y

n+1

= �

h

(y

n

) is a solution of the

multistep method;

� for an arbitrary starting approximation y

0

; : : : ; y

k�1

, the numeri al ap-

proximation of the multistep method tends exponentially fast to a par-

ti ular solution obtained by the underlying one-step method.

The existen e of an underlying one-step method (as a formal series in powers

of h) satisfying the �rst of these properties, an be shown for general on-

sistent methods (2.23); see [12, Chap.XIV℄. The se ond property annot be

ful�lled by methods that are not stri tly stable. Assuming that for arbitrary

starting approximations the multistep solution remains lose to that obtained

by the underlying one-step method, it is natural to all a method (2.23) sym-

ple ti and symmetri , if the underlying one-step method is symple ti and

symmetri , respe tively.

Unfortunately, it turns out that partitioned multistep methods annot

be symple ti (Tang [39℄). However, they an be symmetri . In terms of the

oeÆ ients of the method (2.23), the symmetry of the underlying one-step

method is equivalent to (assuming irredu ibility of the methods)

�

j

= ��

k�j

; �

j

= �

k�j

for j = 0; : : : ; k,

b�

j

= �b�

b

k�j

;

b

�

j

=

b

�

b

k�j

for j = 0; : : : ;

b

k.

(2.24)

For stable symmetri multistep methods the zeros of the generating poly-

nomials �(�) =

P

k

j=0

�

j

�

j

and b�(�) =

P

b

k

j=0

b�

j

�

j

have to lie on the unit


ir le. Su h methods annot be stri tly stable, and for this reason symmetri

multistep methods have been disregarded for a long time.

Also for this lass of methods we are mainly interested in the numeri al

solution of se ond order di�erential equations �q = g(q). Elimination of the

v-variables in (2.23) yields the formula

K

X

j=0

A

j

q

n+j

= h

2

K

X

j=0

B

j

g(q

n+j

); (2.25)

where the generating polynomials R(�) =

P

k

j=0

A

j

�

j

and S(�) =

P

k

j=0

B

j

�

j

are obtained from those of (2.23) by

R(�) = �(�) � b�(�); S(�) = �(�) � b�(�):

Here, �(�); b�(�); �(�); b�(�) are the generating polynomials of �

j

; b�

j

; �

j

;

b

�

j

,

respe tively. We re all that method (2.25) is of order p, if

R(e

h

)� h

2

S(e

h

) = O(h

p+2

) for h! 0: (2.26)

Formula (2.25) does not involve derivative approximations v

n

. If they are

needed, they an be obtained by �nite di�eren es from the position approxi-

mations q

n

.

Example 2.3 (Method Used as `lmm2' in Fig. 2.3). We put K = 8 and we let

R(�) = (� � 1)(�

7

� 1) = (� � 1)

2

(�

6

+ �

5

+ �

4

+ �

3

+ �

2

+ � + 1);

so that all zeros lie on the unite ir le and, apart from � = 1, all zeros are

simple. To get a method of order p = 8, the polynomial S(�) has to satisfy

S(�) = R(�)= log

2

� +O

�

(� � 1)

p

�

( f. ondition (2.26)). Expanding the right-hand expression into a Taylor

series at � = 1 and trun ating to get a polynomial of degree 7, we obtain the

generating polynomial

S(�) =

13207

8640

�

�

7

+ �) �

8934

8640

�

�

6

+ �

2

) +

42873

8640

�

�

5

+ �

3

)�

33812

8640

�

4

:

The resulting method (2.25) is of order 8 for problems �q = g(q), symmet-

ri , and expli it (be ause B

K

= 0). An approximation to the derivative is

obtained by symmetri di�eren es as

_y

n

=

1

840h

�

672 (y

n+1

� y

n�1

)� 168 (y

n+2

� y

n�2

)

+ 32 (y

n+3

� y

n�3

)� 3 (y

n+4

� y

n�4

)

�

:


Composition Methods We onsider the omposition of a given basi one-

step method �

h

(y) with di�erent step sizes:

h

= �

s

h

Æ : : : Æ �

2

h

Æ �

1

h

: (2.27)

The aim is to in rease the order (and hen e the a ura y) while preserving

desirable properties (symple ti ity, symmetry) of the basi method. This idea

has mainly been developed in the papers of Suzuki [38℄, Yoshida [44℄, and

M La hlan [24℄. For a re ent omprehensive survey see [25℄ and Chapters II,

III, and V of [12℄.

The reason of the su ess of omposition methods within geometri inte-

grators are the following properties:

� if �

h

is symple ti , then the omposition method

h

is symple ti ;

� if �

h

is symmetri and if the step size parameters

i

satisfy

i

=

s+1�i

,

then the omposition

h

is symmetri .

The main problem onsists in �nding parameters

i

su h that the omposition

h

is of a given order. Suzuki [38℄ and Yoshida [44℄ propose general simple

pro edures that allow one to onstru t omposition methods of arbitrarily

high order. For orders higher than four they are, however, not very eÆ ient.

One is therefore obliged to investigate and to solve the set of order onditions

for the

i

whi h guarantee that the method

h

of (2.27) has a ertain order.

Example 2.4 (Method Used as ` omp' in Fig. 2.3). From the many published

examples of omposition methods, let us present the oeÆ ients of a method

of order 8 with s = 17 steps:

1

=

17

= 0:13020248308889008087881763

2

=

16

= 0:56116298177510838456196441

3

=

15

= �0:38947496264484728640807860

4

=

14

= 0:15884190655515560089621075

5

=

13

= �0:39590389413323757733623154

6

=

12

= 0:18453964097831570709183254

7

=

11

= 0:25837438768632204729397911

8

=

10

= 0:29501172360931029887096624

9

= �0:60550853383003451169892108

0 1

This set of oeÆ ients is due to Kahan and Li [15℄. The little pi ture to

the right illustrates the 17 steps ne essary for obtaining order 8. The zig-zag

behaviour is typi al for omposition methods. It is impossible to get high

order without negative step sizes.

For the omputations of Fig. 2.3 we use the St�ormer/Verlet s heme (2.12)

as basi integrator. The resulting omposition method is symple ti and (due

to

i

=

18�i

) symmetri .


2.6 Rattle for Constrained Hamiltonian Systems

Let us explain here, how the St�ormer/Verlet method (2.12) an be generalized

to solve onstrained Hamiltonian systems of the form (1.13). Without taking

mu h are of velo ity approximations Ry kaert, Ci otti and Berendsen [31℄

show how onstraints g(q) = 0 an be in luded in the formulation (2.13).

Anderson [1℄ reformulates their method and in ludes a velo ity approxima-

tion that satis�es the hidden onstraint (1.14). The resulting algorithm, still

for separable Hamiltonians, is alled `Rattle'. Later, Jay [16℄ and Rei h [28℄

observed that the Rattle algorithm an be extended to general Hamiltonians.

Re all that the exa t ow of a onstrained Hamiltonian system lies on

the manifold M, de�ned in (1.17). Assume therefore that an approximation

(p

n

; q

n

) 2M is given. One step of the algorithm is de�ned as

p

n+1=2

= p

n

�

h

2

�

r

q

H(p

n+1=2

; q

n

) +r

q

g(q

n

)�

n

�

q

n+1

= q

n

+

h

2

�

r

p

H(p

n+1=2

; q

n

) +r

p

H(p

n+1=2

; q

n+1

)

�

0 = g(q

n+1

)

p

n+1

= p

n+1=2

�

h

2

�

r

q

H(p

n+1=2

; q

n+1

) +r

q

g(q

n+1

)�

n

�

0 = r

q

g(q

n+1

)

T

r

p

H(p

n+1

; q

n+1

):

(2.28)

For �xed �

n

, the �rst two equations de�ne uniquely p

n+1=2

and q

n+1

, if h is

suÆ iently small. The parameter �

n

has to be hosen to satisfy g(q

n+1

) = 0.

This is possible if the matrix (1.16) is invertible. In the last two equations,

�

n

has to be hosen to satisfy the onstraint for p

n+1

.

Similar to the St�ormer/Verlet method for un onstrained Hamiltonian sys-

tems, this algorithm has many ni e properties that are useful within geometri

integration:

� the numeri al solution stays on the manifold M; i.e., the method (2.28)

de�nes a numeri al ow �

h

:M!M;

� the numeri al ow �

h

:M!M is a symple ti transformation on M;

� the method is symmetri ;

� the method is onvergent of order two.

The symple ti ity of the numeri al ow has �rst been shown by Leimkuhler

and Skeel [23℄. The other properties are easy onsequen es of the de�nition

of the method. This integrator is an ideal andidate as basi method for

ompositions of the form (2.27). For elaborate proofs and for extensions to

higher orders we refer to Se t. VII.1 of [12℄.

3 Theoreti al Foundation of Geometri Integrators

Intuitively, it is quite obvious that a symple ti method should be preferred

for the integration of Hamiltonian systems. Similarly, symmetri (more pre-

isely, �-reversible) integrators should be preferred for �-reversible di�erential


equations. This is motivated by the fa t that the symple ti ity of the ow is

hara teristi for Hamiltonian systems, and the �-reversibility of the ow is

hara teristi for �-reversible di�erential equations.

In this se tion we give some more pre ise statements on the long-time be-

haviour of geometri integrators. In parti ular, we dis uss the idea of ba kward

error analysis whi h is the key for a deeper understanding of most numeri-

al phenomena. This idea was ommon to many numeri al analysts already

before a systemati study started with the work of Feng [8℄, Sanz-Serna [33℄,

Yoshida [45℄, Hairer [9℄ and many others.

3.1 Ba kward Error Analysis

Consider an ordinary di�erential equation

_y = f(y) (3.1)

and a numeri al method y

n+1

= �

h

(y

n

). The idea of ba kward error analysis

onsists in sear hing and studying a modi�ed di�erential equation

_y = f(y) + hf

2

(y) + h

2

f

3

(y) + : : : ; (3.2)

su h that the exa t time-h ow e'

h

(y) of (3.2) is equal to the numeri al ow

�

h

(y). Already simple examples (e.g., trapezoidal rule applied to a quadrature

problem _y = f(t)) show that the series in (3.2) annot be expe ted to onverge

in general. The pre ise statement is the following:

Theorem 3.1. Consider the di�erential equation (3.1) with an in�nitely dif-

ferentiable ve tor �eld f(y). Assume that the numeri al ow admits a Taylor

series expansion of the form

�

h

(y) = y + hf(y) + h

2

d

2

(y) + h

3

d

3

(y) + : : : : (3.3)

Then, there exist unique ve tor �elds f

j

(y) su h that for any N � 1

�

h

(y) = e'

h;N

(y) +O(h

N+1

);

where e'

t;N

is the exa t ow of the trun ated modi�ed equation

_y = f(y) + hf

2

(y) + : : :+ h

N�1

f

N

(y)

(noti e that the ow e'

t;N

also depends on h, be ause h is a parameter in the

modi�ed di�erential equation).

Let us outline a onstru tive proof . Without taking are of onvergen e

we expand the exa t ow of (3.2) into a Taylor series

e'

h

(y) = y + h ey

0

(0) +

h

2

2!

ey

00

(0) +

h

3

3!

ey

000

(0) + : : :

= y + h

�

f(y) + hf

2

(y) + h

2

f

3

(y) + : : :

�

+

h

2

2!

�

f

0

(y) + hf

0

2

(y) + : : :

��

f(y) + hf

2

(y) + : : :

�

+ : : :

(3.4)


(where the prime denotes derivation with respe t to time) and ompare like

powers of h in the expressions (3.4) and (3.3). This yields re urren e relations

for the fun tions f

j

(y), namely,

f

2

(y) = d

2

(y)�

1

2!

f

0

f(y) (3.5)

f

3

(y) = d

3

(y)�

1

3!

�

f

00

(f; f)(y) + f

0

f

0

f(y)

�

�

1

2!

�

f

0

f

2

(y) + f

0

2

f(y)

�

:

Example 3.1. We onsider the pendulum equation _q = p; _p = � sin q and

apply the expli it Euler dis retization (2.1). We have d

j

(y) = 0 for all j � 2,

so that (3.5) yields for the modi�ed equation

�

_q

_p

�

=

�

p

� sin q

�

+

h

2

�

sin q

p os q

�

+

h

2

12

�

�4 p osq

(p

2

+ 4 os q) sin q

�

+ : : : : (3.6)

For the impli it Euler method (2.2) we get (3.6) with h repla ed by �h. A

similar omputation yields for the symple ti Euler method (2.3) the modi�ed

di�erential equation

�

_q

_p

�

=

�

p

� sin q

�

+

h

2

�

� sin q

p os q

�

+

h

2

12

�

2 p os q

(p

2

� 2 os q) sin q

�

+ : : : ; (3.7)

whereas the same equation with h repla ed by �h is obtained for the method

(2.4). The four pi tures of Fig. 3.1 show the exa t ow of the modi�ed dif-

ferential equations (trun ated after the O(h

2

) term) orresponding to these

four Euler methods together with the numeri al solution for the initial value

(p

0

; q

0

) = (�1:2; 0:7). We observe a surprisingly good agreement. This �gure

should be ompared to the exa t ow of the unperturbed system ( f. Fig. 1.1).

The O(h) perturbation in (3.6) provokes the origin to be ome a sour e for

the expli it Euler method, and a sink for the impli it Euler method. For the

two symple ti dis retizations we observe that the solutions of the modi�ed

equation are periodi , and that the numeri al approximation lies near a losed

urve. It has thus the orre t qualitative behaviour. This is explained by the

fa t that the di�erential equation (3.7) is Hamiltonian with

e

H(p; q) =

1

2

p

2

� os q �

h

2

p sin q +

h

2

12

(p

2

� os q) os q + : : : ;

so that the exa t solutions stay on the level urves of

e

H(p; q).

3.2 Properties of the Modi�ed Equation

The previous example demonstrates that the numeri al solution is extremely

lose to the exa t solution of the modi�ed di�erential equation. To study

properties of the numeri al solution, it is therefore justi�ed to investigate

instead the orresponding properties of the modi�ed di�erential equation,

and this is often mu h simpler. Let us olle t some properties valid for general

di�erential equations:


−2 0 2 4

−2

2

−2 0 2 4

−2

2

−2 0 2 4

−2

2

−2 0 2 4

−2

2

−2 0 2 4

−2

2

−2 0 2 4

−2

2

−2 0 2 4

−2

2

−2 0 2 4

−2

2

expli it Euler impli it Euler

symple ti Euler (2.3) symple ti Euler (2.4)

Fig. 3.1. Numeri al solution with step size h = 0:4 for the four Èuler methods' of

Se t. 2.1 ompared to the exa t ow of their modi�ed di�erential equations trun-

ated after the O(h

2

) term.

� if the method is of order r, i.e., �

h

(y)� '

h

(y) = O(h

r+1

), then we have

f

j

(y) = 0 for j = 2; : : : ; r ;

� if h

r+1

Æ

r+1

(y) is the leading term of the lo al trun ation error, i.e.,

�

h

(y)�'

h

(y) = h

r+1

Æ

p+1

(y)+O(h

r+2

), then we have f

r+1

(y) = Æ

r+1

(y);

� if �

h

(y) has the modi�ed equation (3.2), then the adjoint method has

f

�

j

(y) = (�1)

j+1

f

j

(y) as oeÆ ient fun tions of the modi�ed equation;

� for symmetri methods the modi�ed equation is an expansion in even

powers of h; i.e., f

2k

(y) = 0 for all k.

We now turn our attention to Hamiltonian systems and to �-reversible dif-

ferential equations.

Theorem 3.2 (Lo al Modi�ed Hamiltonian). Consider a Hamiltonian

system (1.1) with smooth Hamiltonian H : D ! R (D � R

2d

) and apply

a symple ti numeri al method �

h

(y). Then, the ve tor �elds f

k

(y) of the

modi�ed di�erential equation are lo ally Hamiltonian, i.e., lo ally we have

f

k

(y) = J

�1

rH

k

(y).

The proof is by indu tion on k. Its ideas an be tra ed ba k to Moser

[26℄, and it an be found in Benettin & Giorgilli [4℄, Tang [40℄, Rei h [29℄,

and in Chapter IX of [12℄. Sin e the idea of proof is appli able to many other

situations, we outline it shortly.


We assume (by indu tion) that the trun ated modi�ed equation

_y = f(y) + hf

2

(y) + : : :+ h

k�1

f

k

(y) (3.8)

is Hamiltonian. Its ow e'

t;k

(y) satis�es

�

h

(y) = e'

h;k

(y) + h

k+1

f

k+1

(y) +O(h

k+2

):

Sin e �

h

and e'

h;k

are symple ti transformations,

J = �

0

h

(y)

T

J�

0

h

(y) = J + h

k+1

�

f

0

k+1

(y)

T

J + Jf

0

k+1

(y)

�

+O(h

k+2

)

holds. Consequently, the matrix Jf

0

k+1

(y) is symmetri and the existen e of

H

k+1

(y) satisfying f

k+1

(y) = J

�1

rH

k+1

(y) follows from the integrability

lemma. ut

If H and �

h

are both de�ned and smooth on the whole of R

2d

or on a

simply onne ted domain D, the fun tions H

k

of the modi�ed Hamiltonian

are globally de�ned. However, as shown by the following example, the fun -

tions H

k

are in general not globally de�ned, and the above theorem annot

be used for the study of the long-time behaviour of numeri al solutions.

Example 3.2. For the harmoni os illator _p = �q; _q = p, onsider the dis-

retization

p

n+1

= p

n

� hq

n

� h

2

p

n+1

; q

n+1

= q

n

+ hp

n+1

� h

2

q

n

(3.9)

where = 0:25=(p

2

n+1

+q

2

n

). It is aO(h

2

) perturbation of the symple ti Euler

method and therefore it is a method of order 1. Its symple ti ity follows from

the fa t that it an be written as

p

n+1

= p

n

� hr

q

S(p

n+1

; q

n

); q

n+1

= q

n

+ hr

p

S(p

n+1

; q

n

)

with S(p; q) =

1

2

(p

2

+q

2

)�

h

4

arg(q+ip). Its numeri al approximation, plotted

in the right pi ture of Fig. 3.2, is disappointing and does not show the orre t

qualitative behaviour. This is due to the fa t that S(p; q), and hen e H

2

(p; q),

are not globally de�ned.

Theorem 3.3 (Global Modi�ed Hamiltonian). Consider a Hamiltonian

system (1.1) with smooth Hamiltonian H : D ! R (D � R

2d

) and apply the

symple ti method

p

n+1

= p

n

� hr

q

S(p

n+1

; q

n

); q

n+1

= q

n

+ hr

p

S(p

n+1

; q

n

)

with generating fun tion

S(p; q) = S

1

(p; q) + hS

2

(p; q) + h

2

S

3

(p; q) + : : : ;

where all S

k

(p; q) are globally de�ned on D. Then, the ve tor �elds f

k

(y)

of the modi�ed di�erential equation are globally Hamiltonian, i.e., we have

f

k

(y) = J

�1

rH

k

(y) with smooth H

k

: D ! R.


expli it Euler impli it Euler symple ti Eulersympl. meth. (3.9)

Fig. 3.2. Numeri al solution of di�erent �rst order methods applied to the harmoni

os illator with step size h = 0:15.

The proof of this theorem is based on the Hamilton-Ja obi di�erential

equation ( f. Se t. IX.3.2 of [12℄). Let us mention that all previous meth-

ods (symple ti Euler, St�ormer/Verlet, symple ti partitioned Runge-Kutta

methods, omposition methods) satisfy the assumption of Theorem 3.3.

Theorem 3.4 (�-Reversible Modi�ed Ve tor Field). Consider a �-re-

versible di�erential equation ( f. Se t. 1.2) and apply a �-reversible numeri al

method �

h

(y). Then, the ve tor �elds f

k

(y) of the modi�ed di�erential equa-

tion are �-reversible, i.e., they satisfy (1.9).

The proof uses the same ideas as that of Theorem 3.2.

3.3 Long-Time Behaviour of Geometri Integrators

Using ba kward error analysis and in parti ular the results of Theorems 3.3

and 3.4, we shall show that symple ti integrators (for Hamiltonian systems)

and �-reversible integrators (for �-reversible di�erential equations) have an

improved long-time behaviour. We study the onservation of the Hamiltonian

and of general �rst integrals, and the error growth for integrable systems.

Conservation of the Hamiltonian We know that the HamiltonianH(p; q)

is onstant along exa t solutions of the Hamiltonian system (energy onser-

vation for me hani al systems). Sin e the lo al error of a rth order integrator

is of size O(h

r+1

), we have H(p

n+1

; q

n+1

)�H(p

n

; q

n

) = O(h

r+1

). Summing

up these errors, we obtain H(p

n

; q

n

) � H(p

0

; q

0

) = O(nh

r+1

) = O(th

r

) for

t = nh, be ause no an ellation of errors an be expe ted for general integra-

tors. For symple ti integrators, however, we have the mu h more favourable

estimate

H(p

n

; q

n

)�H(p

0

; q

0

) = O(h

r

) for nh � T (3.10)

with an extremely large T (in pra ti e it an be onsidered as in�nity), pro-

vided that the numeri al solution stays in a ompa t set. This an be ex-

plained with the help of Theorem 3.3 as follows: the modi�ed di�erential

equation is Hamiltonian with

e

H(p; q) = H(p; q) + h

r

H

r+1

(p; q) + h

r+1

H

r+2

(p; q) + : : : : (3.11)


The exa t ow of the modi�ed equation, and hen e also the numeri al solu-

tion, keep the modi�ed Hamiltonian

e

H(p; q) exa tly onstant. If the numeri al

solution stays in a ompa t set, the fun tions H

j

(p; q) are bounded along the

numeri al solution so that (3.10) holds. This argument is not yet rigorous,

be ause the series (3.11) usually does not onverge. If one trun ates the series

suitably, one an rigorously prove (3.10) on exponentially long time intervals,

i.e., for T = O(e

=h

) with some positive ( f. [4℄, [11℄, and Se t. IX.7 of [12℄).

It is natural to study whether also other �rst integrals an be well on-

served by numeri al integrators. Re all that I(y) is a �rst integral of _y =

f(y), if it is onstant along all solutions of the di�erential equations, i.e., if

I

0

(y)f(y) = 0 vanishes identi ally.

Example 3.3. Consider the Kepler problem (1.11). Besides the Hamiltonian

(2.17), it has also the angular momentum

L(p

1

; p

2

; q

1

; q

2

) = q

1

p

2

� q

2

p

1

: (3.12)

and the so- alled Runge-Lenz-Pauli ve tor

A(p; q) =

0

�

p

1

p

2

0

1

A

�

0

�

0

0

q

1

p

2

� q

2

p

1

1

A

�

1

p

q

2

1

+ q

2

2

0

�

q

1

q

2

0

1

A

(3.13)

as �rst integrals. As numeri al s heme we take the St�ormer/Verlet method.

We apply it to the Kepler problem with initial values as in Se t. 2.5, and we

use the step size h = 0:02. Figure 3.3 shows the values of H(p

n

; q

n

)�H(p

0

; q

0

)

and of the �rst two omponents of A(p

n

; q

n

)�A(p

0

; q

0

) along the numeri al

solution. The angular momentum L(p; q) is exa tly preserved by the method

and therefore not visible in the �gure. We see that, in agreement with (3.10),

the error in the Hamiltonian is bounded by O(h

2

) on the whole interval of

integration. The Runge-Lenz-Pauli ve tor (3.13), however, is not preserved.

The lower pi ture of Fig. 3.3, where the errors obtained with step size h = 0:04

are in luded in grey, indi ates that they behave like e(h

2

t) +O(h

2

).

Completely Integrable Systems The example above demonstrated that

it is diÆ ult to predi t the onservation of general �rst integrals by numeri al

methods (even when they are symple ti ). There is, however, an important

spe ial ase for whi h more information an be obtained. We mention some

fa ts and refer the reader to Chapters X and XI of [12℄.

We all a Hamiltonian system (1.1) ompletely integrable, if there exists

a symple ti transformation

(p; q) = (a; �); 2�-periodi in �; (3.14)

su h that the Hamiltonian be omes

H(p; q) = H

�

(a; �)

�

= K(a): (3.15)


100 200 300 400

−.004

−.002

.000

.002

.004

5000 10000 15000

−1.0−.5.0.5

1.0

H

A

1

A

2

onservation of �rst integrals

A

1

A

2

Fig. 3.3. Kepler problem: the Hamiltonian and the �rst two omponents of

the Runge-Lenz-Pauli ve tor along the numeri al solution of the St�ormer/Verlet

method with step sizes h = 0:02 (bla k) and h = 0:04 (grey).

The new variables (a; �) are alled a tion-angle variables. Suppose we know

expli itly the transformation . Sin e it is symple ti , the Hamiltonian sys-

tem (1.1) be omes in the new variables

_a

i

= 0;

_

� = !

i

(a); i = 1; : : : ; d

with !

i

(a) = �K=�a

i

(a). This system an be readily solved, and gives a

i

(t) =

a

i0

, �

i

(t) = �

i0

+ !

i

(a

0

)t, so that

�

p(t); q(t)

�

=

�

a

0

; �

0

+ !(a

0

)t

�

:

The gives a periodi or quasi-periodi ow on the torus de�ned by a = onst .

Among the problems seen in this survey arti le, Hamiltonian systems with

one degree of freedom (harmoni os illator, pendulum) and the Kepler prob-

lem are ompletely integrable. Under some additional te hni al assumptions

(see the general referen e [12℄), symple ti numeri al integrators applied to

su h ompletely integrable Hamiltonian systems have the following interest-

ing properties:

� the global error grows at most linearly with time, more pre isely, for

t = nh we have

p

n

� p(t) = O(h

r

t); q

n

� q(t) = O(h

r

t);

� �rst integrals that only depend on the a tion variables are well preserved

on exponentially long time intervals; i.e., if I(p; q) is su h that I( (a; �))

is independent of �, then

I(p

n

; q

n

)� I(p

0

; q

0

) = O(h

r

) for nh � T

with T as in (3.10).


This result has to be seen in ontrast to general methods, where the global

error in reases typi ally quadrati ally with time, and where the error in �rst

integrals drifts linearly from the orre t value.

We �nally mention that the notion of omplete integrability an be re-

interpreted for �-reversible di�erential equations (not ne essarily Hamilto-

nian), and the same results (linear error growth, onservation of a tion vari-

ables) hold for �-reversible integrators applied to su h systems. Let us illus-

trate this at an interesting example.

Example 3.4 (Toda latti e). Let us onsider parti les on a line intera ting

pairwise with exponential for es, and suppose periodi boundary onditions

q

d+1

= q

1

. The Hamiltonian is given by

H(p; q) =

d

X

k=1

�

1

2

p

2

k

+ exp(q

k

� q

k+1

)

�

:

The orresponding Hamiltonian system has the interesting property that the

d eigenvalues of the matrix

L =

0

B

B

B

B

B

B

�

a

1

b

1

b

d

b

1

a

2

b

2

0

b

2

.

.

.

.

.

.

0

.

.

.

a

d�1

b

d�1

b

d

b

d�1

a

d

1

C

C

C

C

C

C

A

;

a

k

= �

1

2

p

k

b

k

=

1

2

exp

�

1

2

(q

k

� q

k+1

)

�

are �rst integrals. This Hamiltonian system is ompletely integrable within

the lass of Hamiltonian systems and also within the lass of �-reversible sys-

tems, and the a tion variables are related to the eigenvalues of the matrix L.

We onsider the ase d = 3, and we apply an impli it Runge-Kutta

method (Lobatto IIIB, s = 3, order r = 4) whi h is symmetri but not

symple ti . The upper pi ture of Fig. 3.4 shows the Eu lidean norm of the

ve tor of errors in the eigenvalues of L. It is of size O(h

4

) on the whole in-

terval of integration. The lower pi ture shows the norm of the global error,

and we ni ely observed the linear error growth. This on�rms the statement

about integrable systems of this se tion.

4 Matlab Programs of `GniCodes'

We explain a few Matlab programs that implement the most important geo-

metri integrators of the previous se tions. They are olle ted in the Matlab

pa kage GniCodes whi h is available (together with short installing instru -

tions) on the web at the address

http://www.unige. h/math/folks/hairer


50 100 150 200.0000.0002.0004.0006.0008

50 100 150 200.0

.2

.4

error in the eigenvalues of L

Lobatto IIIB, h = 0:1

global error

Lobatto IIIB, h = 0:1

Fig. 3.4. The Toda latti e: error in the eigenvalues of L (upper pi ture), and global

error of the numeri al solution for an impli it Runge-Kutta method.

Fortran 77 versions of the programs are also available at the same address.

Another Matlab pa kage related to geometri integration is DiffMan of [7℄.

The philosophy of our pa kage, however, is ompletely di�erent and it is

loser to the standard Matlab ODE suite (ode45, ode23, et ) of Shampine

and Rei helt [34℄.

4.1 Standard Call of Integrators

We give an overview of how to use the three lasses of geometri integrators

that are implemented in the pa kage GniCodes for the moment. For the

solution of se ond order initial value problems

�q = g(q); q(0) = q

0

; _q(0) = _q

0

(4.1)

all these methods have the same syntax, and are usually alled as

[T,P,Q℄ = gni_meth('g',tspan,y0,options,...);

where gni meth has to be repla ed by gni irk2 for the impli it Runge-Kutta

method, by gni lmm2 for the linear multistep method, and by gni omp for

the omposition method based on the St�ormer/Verlet s heme. The `2' in irk2

and in lmm2 expresses the fa t that these programs are appli able only to se -

ond order di�erential equations (4.1). For the omposition method gni omp

there is the possibility to de�ne the basi integrator by altering the options

stru ture, so that the method an be used for the solution of any di�erential

equation. The relevant syntax will be explained in Se t. 4.6 below.

The meaning of the arguments in a all of gni meth is as follows:

'g' This argument must be a string ontaining the name of a Matlab �le

des ribing the problem. The syntax for su h a �le is des ribed in Se t. 4.2

below.


tspan should ontain the time span over whi h the problem is to be solved.

It has to be given in the form [t0,tf℄.

y0 This is a ve tor ontaining the initial values for q and _q. The initial values

for q are given by the �rst d omponents and those for _q by the remaining

d omponents of y0.

options This argument should ontain a GNI options stru ture reated by

gniset (the syntax of gniset is the same as for the standard odeset

fun tion). This option stru ture ontains additional instru tions for the

integrator.

... After options, an arbitrary number of optional arguments an be given.

These arguments are passed over to the fun tion F.

The list of available options di�ers slightly from the standard ODE suite.

Some of these options are also fun tion-dependent and will be explained in

the sequel. The following options are available for every integrator:

'OutputF n' This is a string ontaining the name of an output fun tion. The

format for the output fun tions is the same as for the standard ODE suite,

in parti ular the standard odeplot output fun tion an be used (and will

be used as the default output fun tion). The ve tor passed to the output

fun tion ontains in its d �rst omponents the value of the solution and

in its d remaining omponents the values for its time-derivatives. If this

parameter is set to phaseplot, the solution is drawn in the phase spa e

orresponding to the �rst omponents of (q; _q).

'OutputSel' As for the standard ODE suite, this ontains a ve tor of indi es

determining whi h omponents of the solution will be passed to the out-

put fun tion. By default, all the indi es (in luding those orresponding

to the time-derivative of the solution) are passed through.

'OutputSteps' tells the integrator whi h steps to take into a ount for the

output. For example, if OutputSteps is equal to 10, only every 10th step

generates some output. Putting OutputSteps equal to 0, output is made

available only at the beginning and the end of the integration interval.

'Ve torized' has the same meaning as for the standard ODE suite.

'Events' If this option is set to 'on', event lo ation is enabled. See Se t. 4.3

for an explanation of how to use event lo ation.

'StepSize' Size h of one integration step. It is slightly altered by the ode,

if the length of the integration interval is not an integer multiple of h.

'NumSteps' is the number of integration steps. This option is only used when

StepSize is not spe i�ed. If neither is spe i�ed, a warning is issued and

the default step size h = 0:01 is used.

'Method' allows to sele t the type of method to use. The list of available

methods depends on the integrator and is listed in the orresponding

se tions below.

Note that (like for ode45 for example) the arguments tspan, y0, and options

are optional and an be de�ned in the �le F.m instead.


On output, gni meth returns three ve tors [T,Q,P℄, ontaining the times

at whi h the solution was evaluated, as well as the values of q and _q at these

times. If event lo ation is turned on, additional return values are given as

des ribed in Se t. 4.3 below.

4.2 Problem Des ription

The problem to be solved should be des ribed in a .m �le. In the most simple

ase, this �le only returns the right-hand side of the se ond-order di�erential

equation. E.g., in order to solve the equation �q = �q

3

, one may reate a �le

trivial.m ontaining the following:

fun tion out = trivial(t,q)

out = -q^3;

From the ommand line, one would then use it for example as

options = gniset('StepSize', 0.1);

gni_meth('trivial', [0 10℄, [0 2.5℄, options);

An additional parameter flags an be used by the integrator to retrieve

default parameters for the problem. Assume we want to solve the previous

problem between t = 0 and t = 10, using a step size of 0:1 and with initial

values q(0) = 0 and _q(0) = 2:5. We ould then de�ne the �le trivial.m as

fun tion [out,out2,out3℄ = trivial(t,q,flags)

if (nargin < 3) | isempty(flags)

out = -q^3;

else

swit h flags

ase 'init',

out = [0 10℄;

out2 = [0 2.5℄;

out3 = gniset('StepSize', 0.1);

end

end

and all it from the ommand line in the most simple possible way as

gni_meth('trivial');

For a system of di�erential equations, out and q are olumn ve tors. If the

option 'Ve torized' is set to 'on' in the GNI options stru ture, the inte-

grator may request to evaluate the right-hand side of the problem for several

values of t and q in one all. If the problem is of dimension d and the inte-

grator requests m values, t is a line ve tor of size m and q is a d�m matrix.

The right-hand side is expe ted to be also a d�m matrix. If 'Ve torized'

is set to 'off', one an safely assume that m = 1.


When ve torized orre tly, the .m �le for the Kepler problem with initial

values as in Se t. 2.5 looks like

fun tion [out,out2,out3℄ = kepler(t,q,flags,e )


rad=q(1,:).*q(1,:)+q(2,:).*q(2,:);

rad=rad.*sqrt(rad);

out(1,:)=-q(1,:)./rad;

out(2,:)=-q(2,:)./rad;

else

swit h flags

ase 'init',

if (e < 0) | (e >= 1)

error('The e entri ity must lie between 0 and 1');

end

out = [0 2*pi℄;

out2 = [1-e ,0,0,sqrt((1+e )/(1-e ))℄;

out3 = gniset('NumSteps',50,'Ve torized','on','Events','off',...

'OutputF n','phaseplot','OutputSel',[1,2℄);

end

end

Noti e that this problem depends on the e entri ity e , whi h has been

appended to the end of the parameter list. To solve this problem with e =

0:6, just type

gni_meth('kepler',[ ℄,[ ℄,[ ℄,0.6);

Entering [ ℄ in the parameter list tells the integrator to use the default

values of the problem de�nition �le instead. The parameter e is again

simply appended at the end of the parameter list.

4.3 Event Lo ation

In many situations (for example the omputation of Poin ar�e se tions), it is

useful to know at whi h times some event fun tion g(t; q(t); p(t)) vanishes.

This is usually referred to as event lo ation.

Event lo ation is implemented in the GNI suite in a way that is again

very similar to the standard ODE suite implementation. It an be enabled

by spe ifying the value 'on' for the 'Events' sele tor of the GNI options

stru ture.

When event lo ation is turned on, the integrator an be alled as

[T,P,Q,TE,PE,QE,IE℄ = gni_meth('g',tspan,y0,options,...);

The output ve tor TE ontains the times at whi h events o urred. The ve -

tors PE and QE ontain the values of the solution and its derivative at these

times. If more than one event fun tion is de�ned, the ve tor IE ontains the

index of the event fun tion that triggered the event.


When event lo ation is turned on, the problem des ription �le is expe ted

to respond to the flags set to 'events' by returning in the �rst output

argument a ve tor of event fun tions. Furthermore it is supposed to return in

the se ond and third output arguments ve tors telling the integrator whether

the orresponding event is terminal or not and whi h types of zero- rossings

to onsider. When a terminal event is en ountered, the integration stops,

whether the end of the integration interval has been rea hed or not. The

following example shows how to de�ne a problem des ription �le that allows

to retrieve the times at whi h the solution either rosses 1 upwards or 0 in any

dire tion. The integration stops whenever the solution rosses�2 downwards.

fun tion [out,out2,out3℄ = trivial(t,q,flags)


out = -q^3;

else

swit h flags

ase 'init',

out = [0 10℄;

out2 = [-1 5℄;

out3 = gniset('StepSize', 0.1,'Events','on');

ase 'events',

out = [q(1)-1,q(1),q(1)+2℄;

out2 = [0 0 1℄;

out3 = [1 0 -1℄;

end

end

4.4 Program gni irk2

The program gni irk2 uses an impli it Runge-Kutta s heme to solve se ond

order di�erential equations �q = g(q). The following sele tors of the GNI

options stru ture are spe i� to gni irk2:

'Method' This sele tor allows to spe ify whi h s heme is to be used. The

a epted values for 'Method' are 'G4', 'G8', and 'G12'. The letter `G'

refers to the fa t that all of these methods are Gauss methods ( f. Example

2.2), and the number that follows indi ates the order of the orresponding

method. If no method is spe i�ed, 'G12' is used.

'MaxIter' Sin e the s hemes are impli it, a non-linear system of equation

has to be solved at every integration step. This is a hieved through a

�xed point iteration. This sele tor allows to spe ify the maximal number

of iterations that are performed. The default value is 50.

The oeÆ ients of the di�erent methods are ontained in the �le oeff irk2.

New methods an easily be in orporated.

Let us shortly explain the meaning of the required oeÆ ients. The arrays

C, B, BC orrespond to the ve tors with oeÆ ients

i

,

b

b

i

,

e

b

i

of (2.22), the 2-

dimensional array AA to the matrix

e

A. Further oeÆ ients are needed for an


eÆ ient solution of the nonlinear Runge-Kutta equations of (2.22), whi h are

equivalent to

Q

i

= q

n

+ h

i

v

n

+ h

2

s

X

j=1

ea

ij

g(Q

j

); i = 1; : : : ; s: (4.2)

We solve this system by �xed point iteration and we use

Q

0

i

= q

n

+ h

i

v

n

+ h

2

s+3

X

j=1

e

ij

g(Q

j;n�1

) (4.3)

as starting guess, where Q

1;n�1

; : : : ; Q

s;n�1

are the internal stage values of

the previously omputed step, Q

s+1;n�1

= q

n�1

, Q

s+2;n�1

= q

n

, and

Q

s+3;n�1

= q

n

+

s

X

i=1

�

i

�

Q

i;n�1

� q

n�1

�

+ h�

s+1

v

n�1

+ h�

s+2

v

n

is an approximation to the solution at t = t

n�1

+ �h. The oeÆ ients �,

�

i

, e

ij

(stored in the arrays SM, AM, and E) are determined su h that (4.3)

oin ides as far as possible with the Taylor series of the solution of (4.2). We

refer to [12, Se t. VIII.6.1℄ and [20℄ for more details.

4.5 Program gni lmm2

Linear multistep methods (2.25) for se ond order di�erential equations are

implemented in the ode gni lmm2. Sin e these methods are not self-starting,

we have to provide starting approximations. This is done by a all to gni irk2

with 'Method' set to 'G12'. For the moment we have implemented the three

methods of Table 1, and the 'Method' options are '801', '802', and '803',

respe tively.

Table 1. Symmetri multistep methods for se ond order problems.

method 801 method 802 method 803

i C

i�1

12096B

i

C

i�1

120960B

i

C

i�1

8640B

i

1 1 17671 1 192481 1 13207

2 0 �23622 2 6582 1 �8934

3 1 61449 3 816783 1 42873

4 1 �50516 3:5 �156812 1 �33812

The oeÆ ients of the methods are stored in the separate �le oeff lmm2

as follows. The generating polynomial R(�) has � = 1 as a double zero, and

therefore it an be written as

R(�) = (� � 1)

2

(C

0

+ C

1

� + C

2

�

2

+ : : :+ C

k�2

�

k�2

):


Sin e for expli it symmetri methods these oeÆ ients satisfy B

K�i

= B

i

(with B

0

= 0) and C

K�2�i

= C

i

, only those given in Table 1 have to be

spe i�ed. The oeÆ ients C

i

and B

i

uniquely determine the method (2.25).

4.6 Program gni omp

This program allows to easily implement general omposition methods. A

omposition method (2.27) is hara terized by the set of oeÆ ients f

i

g and

by the basi method �

h

. They are ontrolled by the following two options:

'Method' This option allows to hoose between several prede�ned sets of

oeÆ ients

i

. The available methods are '21', '43', '45', '67', '69',

'815', '817', and '1033'. These methods are of order 2, 4, 6, 8, and 10

respe tively.

'PartialFlow' This option allows the user to spe ify the name of a Matlab

�le de�ning the basi method �

h

. The default method is the St�ormer/Verlet

method.

For reasons of eÆ ien y we assume the basi method to be of the form

�

h

= !

h=2

Æ �

h=2;h;h=2

Æ �

h=2

; (4.4)

where the following simpli� ation formula holds:

�

�;�;ha

Æ �

ha

Æ !

h

Æ �

h ;�;�

= �

�;�;ha+h

Æ �

ha+h ;�;�

: (4.5)

Every one-step method an be written in the form (4.4) by hoosing !

h=2

=

�

h=2

= id (the identity), and �

h ;hb;ha

= �

hb

. But this is not the reason for

writing �

h

in this apparently ompli ated form. The advantage of the repre-

sentation (4.4) is that in many important situations a large part of the work

for evaluating �

h

an be put into �

h=2

and !

h=2

and, by the simpli� ation

formula (4.5), this part an be avoided unless at grid points where an output

of the solution is required.

The ode for the basi method must have the following stru ture:

fun tion [outP,outQ℄ = basi (t,P,Q,ode,ha,hb,h ,first,last,flags,args)

if isempty(flags)

if (first)

apply �

ha

to the ve tors P and Q

end

apply �

h ;hb;ha

if (last)

apply !

h

end

else

swit h flags

ase 'init',

perform some initialization

ase 'done',


perform some leanup

end

end

For example, the St�ormer/Verlet method (2.12) for �q = g(q), onsidered as a

splitting method (2.16), an be written as (4.4) with !

h=2

= id, �

h ;hb;ha

=

'

[1℄

h

Æ'

[2℄

hb

, and �

ha

= '

[1℄

ha

. This presentation satis�es ondition (4.5), be ause

'

[1℄

t

has the group property. The Matlab program for this basi method is

fun tion [outP,outQ℄ = stverl(t,P,Q,ode,ha,hb,h ,first,...

last,flags,varargin)

if isempty(flags)

if (first)

Q = Q + ha*P;

end

F = feval(ode,t,Q,varargin{:});

outP = P + hb*F;

outQ = Q + h *outP;

end

The a tual implementation uses ompensated summation (to redu e round-

o� error) and it is the default method used by gni omp.

5 Some Typi al Appli ations

Let us �nally illustrate the use of our programs at some typi al examples,

where geometri integrators are re ommended. We start with a omparison

of geometri integrators for se ond order ordinary di�erential equations. We

then show how Poin ar�e se tions an be omputed, and we terminate with a

slightly more sophisti ated use of omposition methods.

5.1 Comparison of Geometri Integrators

Often it is diÆ ult to de ide whi h integrator is the best for a given problem.

The impli it Runge-Kutta and omposition methods have a sound theoreti al

basis, but they typi ally need more fun tion evaluations per step than linear

multistep methods. On the other hand, multistep methods have larger lo al

error, so that smaller step sizes are required. The best hoi e is in general

problem dependent.

Consider �rst the Kepler problem with e entri ity e = 0:6. The �le

kepler.m, ontaining the problem des ription, is explained in Se t. 4.2. We

ompute the solution over the interval [0; 400�℄ with many di�erent step sizes.

As we have seen in Fig. 2.3 the eÆ ien y of all three lasses of integrators

(impli it Runge-Kutta, multistep, omposition) is about the same for this

problem. This need not always be the ase.

As another example onsider the 6-body problem (sun and the �ve outer

planets) with data and initial values as in Chap. I of [12℄ on a relatively short


time interval [0; 500 000℄. Similar as in Fig. 2.3 we show in Fig. 5.1 the work

pre ision diagram of the di�erent methods. It is somewhat surprising that

for this problem (with orbits of very small e entri ity) the linear multistep

method is the most eÆ ient integrator. The pu times in Fig. 2.3 and in

Fig. 5.1 are obtained with Fortran implementations of the odes.

5.2 Computation of Poin ar�e Se tions

Consider the H�enon-Heiles Hamiltonian

H(p

1

; p

2

; q

1

; q

2

) =

1

2

(p

2

1

+ p

2

2

) +

1

2

(q

2

1

+ q

2

2

) + q

2

1

q

2

�

1

3

q

3

2

: (5.1)

The orresponding Hamiltonian system is integrable for suÆ iently small

energy, e.g., for the initial values p

1

(0) = p

2

(0) = q

1

(0) = q

2

(0) = 0:18,

whi h we take for our omputations. This means that the solution stays on

a two-dimensional torus in the four-dimensional phase spa e. Its interse tion

with the hyperplane q

1

= 0 (Poin ar�e se tion) thus gives a losed urve in the

phase spa e. We study the proje tion of this urve onto the (q

2

; p

2

)-plane.

The left pi ture of Fig. 5.2 shows the Poin ar�e se tion for the numeri al

solution obtained by dop853 (an expli it Runge-Kutta method of order eight

with step size ontrol, see [14, Appendix℄) and toleran e Atol = Rtol = 10

�5

on the interval [0; 100 000℄. The pi ture learly demonstrates that the nu-

meri al solution is qualitatively wrong as it does not remain on a losed

urve. The same experiment with the three geometri integrators gni lmm2,

gni irk2, and gni omp gives a orre t simulation of the system, and it an-

not be distinguished from a pi ture for the exa t solution (right pi ture of

Fig. 5.2). If we use step sizes su h that the error of the Hamiltonian remains

104 105

10−9

10−6

10−3

100

10−1 100

10−9

10−6

10−3

100

e

r

r

o

r

f n. eval.

omp

irk2

lmm2

verlet

e

r

r

o

r

pu time

omp

irk2

lmm2

verlet

Fig. 5.1. Work pre ision diagrams for the St�ormer/Verlet s heme and for three

methods of order eight; impli it Runge-Kutta method (irk2), omposition method

( omp), and linear multistep method (lmm2), applied to the outer solar system.


below 10

�5

, the ode gni lmm2 (with h = 0:22) requires 454 716 fun tion

evaluations, gni irk2 (with h = 1:5) needs 3 731 867 fun tion evaluations,

and gni omp (with h = 1:2) 1 416 661 fun tion evaluations. For omparison,

the ode dop853 requires 1 216 680 evaluations of the ve tor �eld, but the

error in the Hamiltonian in reases linearly with time. The high number of

fun tion evaluations for gni irk2 is due to the fa t that for low a ura y

requirements (large step size) the onvergen e of the �xed point iterations

for solving the nonlinear Runge-Kutta equations is rather slow.

For the omputation of the Poin ar�e se tion (Fig. 5.2) we have used the

following program:

fun tion [out,out2,out3℄ = henon(t,q,flags)


out(1,:)=-q(1,:).*(1+2*q(2,:));

out(2,:)=-q(2,:).*(1-q(2,:)) - q(1,:).^2;

else

swit h flags

ase 'init',

out = [0 100000℄;

out2 = [0.18 0.18 0.18 0.18℄;

out3 = gniset('StepSize',0.22,'Ve torized','on',...

'Events','on','OutputSteps',0);

ase 'events',

out = [q(1)℄;

out2 = [0℄;

out3 = [0℄;

end

end

dop853 gni lmm2

Fig. 5.2. Poin ar�e se tion for the H�enon-Heiles problem: dop853 with toleran e

Atol = Rtol = 10

�5

(left pi ture) and gni lmm2 with step size h = 0:22 (right

pi ture); integration interval [0; 100 000℄.


The plot of the Poin ar�e se tion is then obtained with

[T,Q,P,TE,QE,PE℄=gni_lmm2('henon');

plot(QE(:,2),PE(:,2),'.');

5.3 `Rattle' as Basi Integrator for Composition

As a �nal example, we present a Matlab implementation of the Rattle algo-

rithm (2.28) applied to the two-body problem on the sphere as introdu ed in

Example 1.4. We follow the des ription of Se t. 4.6 and we do it in su h a way

that it an be used as basi integrator for the omposition method gni omp.

A possible implementation is the following program:

fun tion [outP,outQ℄ = rattwo(t,P,Q,gradpot,ha,hb,h ,first,last,...

flags,varargin)

if isempty(flags)

F = feval(gradpot,t,Q,varargin{:});

EP = P - ha*F;

EQ = Q + hb*EP;

EE1 = EQ(1:3)'*EQ(1:3);

EQ1 = EQ(1:3)'*Q(1:3);

EE2 = EQ(4:6)'*EQ(4:6);

EQ2 = EQ(4:6)'*Q(4:6);

BET1 = 1 - EE1;

ALAM1 = -BET1/(hb*(EQ1+sqrt(BET1+EQ1^2)));

BET2 = 1 - EE2;

ALAM2 = -BET2/(hb*(EQ2+sqrt(BET2+EQ2^2)));

outP = EP - [ALAM1*Q(1:3);ALAM2*Q(4:6)℄;

outQ = Q + hb*outP;

if (last)

F = feval(gradpot,t,outQ,varargin{:});

outP = outP - h *F;

AMU1 = sum(outP(1:3).*outQ(1:3));

AMU2 = sum(outP(4:6).*outQ(4:6));

outP = outP - [AMU1*outQ(1:3);AMU2*outQ(4:6)℄;

end

end

We remark that, due to the simple stru ture of the Hamiltonian, the method

is expli it in p

n+1=2

, q

n+1

and p

n+1

, and it is impli it only in the Lagrange

multipliers. Sin e the onstraints are quadrati , we are only on erned with

the solution of a s alar quadrati equation for ea h of the omponents of �

n

.

This is why no iterations are involved in the above program. Sin e r

q

H(p; q)

does not depend on p, the �rst equation of (2.28) an be ombined with the

fourth equation of the pre eeding step into one formula to give p

n+1=2

=

p

n�1=2

+ : : : . This is the reason for putting the omputation of p

n+1

into

!

h=2

of the de omposition (4.4).


500 1000 1500

−.000008

−.000004

.000000

.000004

Fig. 5.3. Error in the Hamiltonian of an 8th order omposition method with Rattle

as basi integrator; step size h = 0:15

The argument gradpot in the fun tion rattwo is a fun tion that omputes

the gradient of the potential (i.e., r

q

U(q) = r

q

H(p; q)). For the two-body

problem on the sphere it is given by

fun tion [out,out2,out3℄ = twobodysphere(t,q,flags)


prod = q(1:3)'*q(4:6);

out = -q([4:6,1:3℄)/(1-prod^2)^(3/2);

else

swit h flags

ase 'init',

out = [0 10℄;

phi = [1.3 -2.1℄;

theta = [2.1 -1.1℄;

out2([1 4℄) = os(phi).*sin(theta);

out2([2 5℄) = sin(phi).*sin(theta);

out2([3 6℄) = os(theta);

dphi = [1.2 0.1℄;

dtheta = [0.1 -0.5℄;

out2([7 10℄) = -dphi.*sin(phi).*sin(theta) ...

+ dtheta.* os(phi).* os(theta);

out2([8 11℄) = dphi.* os(phi).*sin(theta) ...

+ dtheta.*sin(phi).* os(theta);

out2([9 12℄) = -dtheta.*sin(theta);

out3 = gniset('StepSize',0.02,'Ve torized','off',...

'Events','off','PartialFlow','rattwo','OutputF n',...

'sphereplot','OutputSteps',5,'Method','817');

end

end

Here, the option sphereplot permits us to get a 3-dimensional plot of the

solution. The problem is then simply solved by alling

gni_ omp('twobodysphere');

The experiment of Fig. 5.3 on�rms the statement of Theorem 3.3 for the

Rattle algorithm applied to onstrained Hamiltonian systems. We have plot-


ted the error of the Hamiltonian for the omposition method of Example 2.4,

applied with step size h = 0:15 to the interval [0; 2000℄. As expe ted for a

symple ti integrator, there is no drift in the error of the Hamiltonian. This

is also on�rmed by integrations over mu h longer time intervals.

Hints for the implementation of the rigid body problem of Example 1.5 an

be found in [10℄, where also the extension of Theorem 3.3 to numeri al meth-

ods for Hamiltonian systems on manifolds (in luding the Rattle algorithm)

is proved.

A knowledgement

E.H. is grateful to the organizers of the 2002 Durham summer s hool, in

parti ular to James F. Blowey, to Alan W. Craig, and to the lo al expert

Sebastian Rei h, for providing a sympathi al and stimulating athmosphere.

We also wish to thank Christian Lubi h, Gerhard Wanner and numerous

other olleagues for their useful dis ussions on the topi of this work.

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