SIAM J. NUMER. ANAL.Vol. 33, No. 1, pp. 368-387, February 1996

() 1996 Society for Industrial and Applied Mathematics019

SYMPLECTIC PARTITIONED RUNGE-KUTTA METHODS FOR CONSTRAINEDHAMILTONIAN SYSTEMS*

LAURENT JAY

Abstract. This article deals with the numerical treatment of Hamiltonian systems with holonomic constraints.A class of partitioned Runge-Kutta methods, consisting of the couples of s-stage Lobatto IIIA and Lobatto IIIBmethods, has been discovered to solve these problems efficiently. These methods are symplectic, preserve all un-

derlying constraints, and are superconvergent with order 2s- 2. For separable Hamiltonians of the form H(q, p)

1/2 prM- p+ U(q) the Rattle algorithm based on the Verlet method was up to now the only known symplectic methodpreserving the constraints. In fact this method turns out to be equivalent to the 2-stage Lobatto IIIA-IIIB method oforder 2. Numerical examples have been performed which illustrate the theoretical results.

Key words, differential-algebraic equations, Hamiltonian systems, index 3, Lobatto methods, mechanical sys-tems, partitioned Runge-Kutta methods, symplecticity

AMS subject classifications. 65L06

1. Introduction. Hamiltonian problems arise in a lot of applications where dissipativeforces can be neglected, such as mechanical systems, astronomy, electrodynamics, moleculardynamics, plasma physics, fluid dynamics, etc. The Hamiltonian system of differential equa-tions associated with the Hamiltonian H(q, p) (a real function supposed sufficiently smooth)is given by

(1.1) 4 Hp (q, P), P -Hqr (q, P)

where q (ql qn)T E R are the generalized coordinates and p (pl pn)r E Rn

the generalized momenta. The flow generated in the phase space R R of (q, p) by theseequations (1.1) is known to be symplectic, i.e., the differential 2-form

(1.2) 092 dq A dp is preserved,

k=l

implying that all differential 2d-forms

(1.3) 092/.. ./NO)2 for d 1,..., n

d times

are also conserved (d n corresponds to the 2n-form volume). Another specific feature ofsuch systems is that the Hamiltonian along a solution (q(t), p(t)) to (1.1) passing through(q0, P0) at to remains invariant, i.e.,

(1.4) H(q(t), p(t)) H(qo, Po) for all t.

Hamiltonian systems also possess numerous other specific properties (see [Ar89, Part III] and[MK92]). Unfortunately, most numerical methods applied to (1.1) do not maintain the abovetwo properties (1.2) and (1.4). Various authors ([SS88], [LAB8], [Sur89], [Y90], [Sun92],[Sun93], [ASS93] among others) have identified or constructed symplectic schemes, i.e.,methods maintaining (1.2). For an overview on symplectic integrators we refer to [SS92]and [HNW93, II. 16].

*Received by the editors June 15, 1993; accepted for publication (in revised form) May 10, 1994.tDepartment of Computer Science, University of Minnesota, 4-192 EE/CS Bldg., 200 Union Street S.E., Min-

neapolis, MN 55455-0159 (na. i jay@na-net, ornl. gov).

368

PRK METHODS FOR CONSTRAINED HAMILTONIAN SYSTEMS 369

In this article we consider Hamiltonian systems with holonomic constraints. Such prob-lems form a particular class of semi-explicit differential-algebraic equations(DAEs) of index 3. We present a very efficient class of partitioned Runge-Kutta (PRK)methods for the solution of these problems. It consists of the couples of s-stage Lobatto Iliaand Lobatto IIIB methods. These methods combine three attractive properties:

--symplecticity, as seen with the RK methods Gauss and Lobatto IllS (see [Sun92],[Sun93], and [Cha90]);

--the fact that the numerical solution can be naturally projected onto the manifolds wherethe exact solution lies (see [AP91], lAP92], [HW91, VI.7], [J94], and [LS94] for similarideas), without loss of symplecticity;

--superconvergence, a property shared by stiffly accurate RK methods such as Radau IIAand Lobatto IIIC (see [HLR89, pp. 18-19, 86], [J93b], [HJ93], [J94], and [J95]).

The importance of symplecticity in numerical integration, especially for long-time compu-tations, is nowadays underlined by a sort of"backward analysis" by interpreting the numericalsolution as the exact solution of a nearby perturbed Hamiltonian system (see [SS92]). In[H94] Hairer recently proved the general result that all symplectic methods whose numericalsolution is a (partitioned) P-series possess this property. An extension of this result to Hamil-tonian systems with holonomic constraints and the numerical methods treated in this articleis likely. This fact is corroborated by the numerical observations of 5.

A typical example of a constrained Hamiltonian system is given by the pendulum equa-tions. Using the cartesian coordinates q (x, z) r for the description of the position of thependulum, the holonomic constraint on the length of the rod of the pendulum is

(1.5) 0-- v/x2+z2-.

The kinetic energy T and the potential energy U of the system are given, respectively, by

rn (32__2) U(q) mgz(1.6) T(4 -where g is the gravitational constant. The Lagrangian of the system is L(q, ) T()-U(q)and the generalized momenta are p (px, pz)r r-LO (q, gt) leading to

(1.7) px m2, Pz mThe Hamiltonian H T+U can be expressed by

1( 2 2)H(q, p) -m Px+P +mgz,

and the Hamilton equations of motion become

Px Pz X Z(1.9) 2 --, i --, /x ) /Sz =-mg--).

rn rn - g

One differentiation of (1.5) implies that

(1.10) 0 Xpx +ZPz

and another one permits us to obtain

(1.11) ;k - (p2x + p2z ) --mgz

370 LAURENT JAY

Thus the differential-algebraic system (1.5) with (1.9) is of differential index 3 (see [BCP89,Chap. 2], [HLR89, p. ], and [HW91, VI.5] for various index definitions).

This paper is organized in five sections. In 2 we give some basic definitions and resultsrelated to symplectic PRK methods. Section 3 deals with the application of PRK methods toHamiltonian systems with holonomic constraints. Convergence results for a specific class ofPRK methods, comprising the symplectic Lobatto IIIA-IIIB schemes, are then stated in 4.Finally, 5 includes some numerical experiments illustrating the theoretical results.

2. Sympleetie PRK methods for Hamiltonian systems. Hamiltonian systems (1.1) areintrinsically split into two parts; therefore the use of PRK methods is very natural.

DEFINITION 2.1. One step ofan s-stage PRK method applied to (1.1), with stepsize h andinitial values (qo, Po) at to reads

(2.1a) ql --qo+h biki Pl po+h ’ii=1 i=1

where

(2.1b) ki H(Qi, Pi) g-i -H(Qi, Pi)

and the internal stages are given by

(2. lc) Qi qo+h aijkj Pi po-b-h aijgjj=l j--1

From now on we use the notation (A b, c) for the RK coefficients where A "-(aij)i,j=l’

b (bl bs) T, c "= (cl Cs) T, and similarly for (, ’,’) (the coefficients ci andienter into the definition of PRK methods for nonautonomous problems).

For a PRK method the symplecticity condition (1.2) is expressed by

(2.2) dqf / dp dq / dpk=l k=l

and symplectic PRK schemes can be characterized as follows:THEOREM 2.1 [H94]. If the coefficients ofan s-stage PRK method (2.1) satisfy

(2.3a) bi bi for s,

(2.3b) biij +bjaji-bibj 0 for s, j s,

then the PRK method is symplectic.If the PRK method is irreducible, then the conditions (2.3) are also necessary for sym-

plecticity.Remarks.(1) For separable Hamiltonians H(q, p) T(p)+U(q), the first condition (2.3a) can

be omitted (see [ASS93]).(2) For a proof of the sufficiency of the conditions (2.3) see [SS88], [La88], [Sur89],

and [Sun92].(3) For irreducible PRK methods, i.e., methods without equivalent stages (see [Bu87,

383], [H94, 4], or the definition of S-irreducibility in [HW91, p. 200]), a way of showingthe necessity of (2.3) is to extend the characterization of canonical B-series of [CSS94] to

PRK METHODS FOR CONSTRAINED HAMILTONIAN SYSTEMS 371

(partitioned) P-series (see [H94, Lemma 11]) and to apply the proof of [H94, Thm. 5]. Forseparable Hamiltonians an alternative proof of the necessity of (2.3b) is given in [ASS93].

DEFINITION 2.2. The local error ofa PRK method (2.1) is defined by

(2.4) qh(tO) ql-q(to+h) ph(tO) pl-p(to+h)

where (q(t), p(t)) is the exact solution of (1.1) passing through (qo, Po) at to.Considering the elegant W-transformation of Hairer and Wanner (see [HW91, IV.5]), it

is possible to construct high order symplectic PRK methods starting from known RK methodsas follows.

THEOREM 2.2 [Sun92]. Suppose that a RK method with coefficients aij, bi O, anddistinct ci, satisfies thefollowing simplifying assumptions:

B(p) bic_ _1 for k p,k

i=1

s, q,k

j-----1

O(r) bic_ bj (1aij - --c)) for j 1 s, k 1 ri--1

then the PRK method (2.1) with coefficients bi bi, i ci, and aij bj (1 aji/bi) issymplectic and satisfies

(2.5) qh(tO) O(h+l), ph(tO) O(h+1)

with an order min(p, 2q +2, 2r +2, q +r + 1).Remarks.(1) With the h..elp of the W-transformation it can be shown that the RK method (A, b, c)

satisfies (r) and D(q) (see [HW91, IV.5] and [Sun92]) where the notation (’), (7") isself-evident.

(2) The simplifying assumptions C (1) and D(1) (which are equivalent to C (1) by thesymplecticity conditions (2.3)) ensure here that ci -j=l aij and ci -j=l aij, respectively.This implies some simplifications when deriving the order conditions of PRK methods ap-plied to nonautonomous problems: in this case the order conditions reduce to those of theautonomous case (see also [HNW93, p. 134]). An example of a RK method violating one ofthese assumptions is given by the 2-stage Lobatto IIIA method, namely the trapezoidal rule,which satisfies B(2), C(2), but not D(1). Another example consists in its dual symplecticmethod, the 2-stage Lobatto IIIB method, which satisfies B(2), D(2), but not C(1).

(3) The symplecticity conditions (2.3), acting as simplifying assumptions, introduce areduction of the number of order conditions (see [ASS93]).

Examples of symplectic PRK methods are given in [Sun92]. In this paper we focus ourattention on PRK methods adapted to the situation where holonomic constraints are encoun-tered. In this context the couples of s-stage Lobatto IIIA methods for (A, b, c) and LobattoIIIB methods for (A, b, c) turn out to be of main interest. These PRK methods satisfy thesimplifying assumptions B(2s-2), C(s), D(s-2), C(s-2), and D(s). Concerning the coeffi-cients of these methods, the weights ci of Lobatto quadratures are given by Cl =0, Cs 1, andthe remaining ci for i- 2 s-1 are the roots of the polynomial of degree s-2 pl,_ (2x-1)where

(2.6) Pl’21 (y) Constds-2

(y2_ 1) dys-((y:- 1)s-l)

372 LAURENT JAY

TABLE 2.1Coefficients of the 2-stage Lobatto IIIA-IIIB method of order 2.

0 0 0 0 1/2 0

1/2 1/2 1/2 0

1/2 1/2 1/2 1/2

1/2

TABLE 2.2Coefficients of the 3-stage Lobatto IIIA-IIIB method oforder 4.

0 0 0 0

5/24 1/3 -1/24 1/2

1/6 2/3 1/6

1/6 2/3 1/6

1/6 -1/6 0

1/6 1/3 0

1/6 5/6 0

1/6 2/3 1/6

is a Jacobi polynomial. The coefficients bj asj and aij can be computed, for example, bythe use of C(s), and the coefficients aj, as in Theorem 2.2 or with the help of (s). TheButcher tableaux of the 2- and 3-stage Lobatto IIIA-IIIB methods are given in Tables 2.1 and2.2, respectively. For separable Hamiltonians the 2-stage method can be applied explicitly.The coefficients of the 4-stage Lobatto IIIA-IIIB method can be found in [HW91, p. 80]. Welist also below the weights ci of the 11-stage Lobatto method of order 20:

(2.7)

1 1Cl 0, C2-" --2, C3--- ’--3, C4 --4, C5-- --5, C6--"

2 Z Z Z

(2 (--u +al)+a3, 3 " - (--u --al +a3

4 (u+a2)+a3, (5 " - u a2 -+-a3

9al /v-4ot, a2 V/v-4fl a3 ot dl-d2 /3 dl +d219’

1080 96912dl= =(el+v), d2 -zv/(el +v)Z-4e2, el e26137’Z Z-- 28800941

32 (1 (2@))720 14132u=/-, v=323c

cos 3arccs -t 6137’c

Due to their symmetry, Lobatto schemes are often used for the solution of boundary valueproblems (see [A85] and [AMR88]). The analysis of the application of Lobatto IIIA methodsto semi-explicit index-2 DAEs in Hessenberg form is given in [J93a].

PRK METHODS FOR CONSTRAINED HAMILTONIAN SYSTEMS 373

3. Hamiltonian systems with holonomic constraints and PRK methods. Mechanicalsystems where dissipative forces can be neglected can be expressed as Hamiltonian systems.Their Hamiltonian is of the form H T + U where T represents the kinetic energy and Uthe potential energy. Usually the equations of motion are not written with a Hamiltonianformalism, but in an Euler-Lagrange formulation (see [HWgl, VI.5 and VI.9]). Our aimhere is to study Hamiltonian systems with m < n holonomic constraints gi (q) 0 (i

m) (see [Cho92, Chap. 2] or JAr89, 17]). A Lagrange-type variational principleexists if the constraints are holonomic (see [Cho92, 4.2.2]). Applying this principle to(q, p)= H(q, p)+)Tg(q), we arrive at

(3. la,b,c) gl H (q, p) D -Hf (q, p)-Gr (q)) 0 g(q)

where G(q) := gq(q). The variables )i (i 1 m) are the Lagrange multipliers. Differ-entiating (3.1c) twice, as in the example given in the Introduction, we obtain the followingadditional constraints (omitting the obvious function arguments):

(3.1d,e) 0- an 0-- aq(n, nTp >f-anTpqn-anpn--anTppaT)Initial values for the problem (3. la,b,c) have to be consistent, i.e., they must satisfy (3. lc,d,e).From now on we suppose that G is of full row rank m and that we have an optical Hamiltoniansystem (see [MK92, p. 140]), meaning that Hpp is a strictly positive definite matrix. Fromthese hypotheses it follows that the matrix GHpGr is invertible; hence we get from (3.1e)

(3.1f) )(q, p) (GHpGT)-1 (Gq(HTp H)nt-GHqHTp -GHpH) (q, p)

and thus the original system (3. la,b,c) is of differential index 3. This explicit relation (3.1f) for) introduced in (3. lb) defines the standard underlying ODE (3.1 a,b) which is not a Hamiltoniansystem in general. All equations (3.1a,b,c,d,e) form an overdetermined system of differential-algebraic equations (ODAEs) (see [FL89] and [PP92]) of index 1. A standard analysis showsthat on the 2(n-m)-dimensional manifold

(3.2) V {(q, p) 6 RnxRn 0--- g(q), 0 G(q)Uf(q, p)}

the flow generated by the equations (3.1) is symplectic (see also [HNW93, I.14]).Disregarding the property (1.4), the ideal properties for a numerical method would be to

be symplectic, to have a numerical solution remaining on the manifold V, and to have a highorder ofconvergence occuring with minimal computational work. The Gauss methods appliedto (3.1a,b,c) are symplectic, but they have the disadvantages that the numerical solution doesnot satisfy the constraints (3.1c,d,e) and that a poor (or even no) convergence occurs (see[HJ93]). Even if projections are effected they are not superconvergent, and the symplecticityproperty is destroyed. To our knowledge the Rattle algorithm, a method of order 2 dueto Andersen (see [An83]) and based on the Verlet method, is the only known symplecticmethod preserving the constraints which has been proposed in the literature for separable

lprM-1Hamiltonians of the form H(q, p) p+U(q) with M a constant positive definitematrix (see [LS94] and [SBO92]). For such Hamiltonians two different approaches have alsobeen derived in [LR94]. The first one is the reduction of (3.1 a,b,c) to a Hamiltonian state-spaceform via a parametrization of the constraints (3.1c). The second one is the construction ofan unconstrained Hamiltonian system which preserves the constraint manifold V and whoseflow reduces to the flow of (3.1) along this manifold. We also mention the independent andsimultaneous investigation by Reich on constrained Hamiltonian systems in [R93] where highorder symplectic methods preserving the constraints are constructed by composition of a first

374 LAURENT JAY

order method (see also [Y90]). For short-time computations, a nonsymplectic alternative is tointegrate the standard underlying ODE and to frequently project the numerical solution ontothe manifolds (3.1c,d,e) (see also lAP93]).

Here we turn our interest to PRK methods.DEFINITION 3.1. The application of an s-stage PRK method to the equations (3.1a,b,c)

reads

(3.3a)

where

ql qo+h biki Pl po+h -ig.ii=1 i=1

ki nTp (Qi, ei),(3.3b)

and the internal stages are given by

(3.3c) Oi qo+h aijkjj=l

ei -nf (ai Pi)--GT (Qi)Ai

Pi po+h _f,.jgjj=l

0-- g(Qi)

i=1 j=l i,j=l

We then replace the differentials dq, dp with the help of (3.3c) and we obtain

(3.5)dq / dp-dqo A dp

h bidkki / deik--lt-h --jdQ) / d.-h2 (biij---jaji-bij)dkki / d.i=1 j=l i,j=l

An easy calculation shows that

(3.6) dk A dP?+ dQ de =- (Oi)dagt 0 dai:k=l k=l 1=1 k=l

An alternative way to understand this foula is as follows: if the variable would be con-stant then as for unconstrained Hamiltonian systems (see [HNW93, Foula II.16.18]) theexpression on the left-hand side of (3.6) would vanish; hence only the tes involving dAhave to be considered. From the hypotheses (2.3) and the foulas (3.5) and (3.6) we get

(3.7) dqf dp- dq dp --h gi (Qi)dQf m dak=l k=l i=1 I=1 k=l

Remark. The existence and uniqueness of a solution to these equations is not guaranteedwithout some assumptions on the coefficients.

Let us start by supposing that we have a locally unique solution to this system (3.3).Our aim now is to check the symplecticity condition (2.2) for PRK methods with coefficientssatisfying (2.3). It is no surprise that we have the following result.

THEOREM 3.1. If the coefficients of the PRK method (3.3) satisfy (2.3), and if (ql, Pa)are uniquely determined, then the numericalflow (qo, Po) (ql, pl) is symplectic.

Proof. This proof is inspired by the calculations of [SS88], [SS92], and [HNW93,Thm. II.16.6]. We neglect the equations 0 g(Qi) for the moment. Using (3.3a) and thebilinearity of the wedge product "/" we compute for k n

PRK METHODS FOR CONSTRAINED HAMILTONIAN SYSTEMS 375

Now, by the use of g(Qi)-O, we have G(Qi)dQi --0, i.e.,

(3.8)Og

k=l-q (Qi)d Qi o,

which finally gives the desired result. q

This result is another motivation to consider the constraints (3.1c) of index 3 and not thoseof reduced indices (3.1d,e) in Definition 3.1.

RK methods are special cases of PRK methods with coefficients satisfying a’j aij,

bi -bi, andS/-ci. In [J93b], [HJ93], [J94], and [J95], the convergence behavior ofcollocationand RK methods applied to semi-explicit index-3 DAEs in Hessenberg form has been analyzedin detail, confirming the conjecture of [HLR89, p. 86]. Compared to other methods requiringequivalent work, stiffly accurate RK methods, i.e., methods which satisfy

(3.9) asj bj for j s,

are tuned to give highly accurate results when applied to DAEs. Unfortunately this laterassumption and the symplecticity condition biaij -+-bjaji-bibj --0 lead to bs =0 and ais --0for satisfying bi 7 O. Therefore we have the following result.

THEOREM 3.2. No symplectic and stiffly accurate RK schemes exist.This negative result is another motivation for the consideration of PRK methods. For

RK methods with an invertible RK matrix A, one can also easily show that (3.9) implies thatR (oc) --0 where R is the stability function of the method, whereas symplectic schemes mustsatisfy R (ec) 1.

For PRK methods the stiff accuracy condition (3.9) implies that ql Qs and g(ql)g(Qs) 0. For symplectic PRK methods (see (2.3)) satisfying bi 7/= 0 this condition (3.9)implies that

(3.10) ajs=O for j-1 s,

and conversely. Consequently, we now restrict our analysis to PRK methods satisfying (3.10).Under this assumption, As does not influence the solution of the following nonlinear systemoriginating from (3.3b,c):

(3.11)

Oi qo+h --aijHTp (Qj, Pj)j=l

sml

ei Po -h Zaj(H(Qj, Pj)+GV(Qj)Aj)j=l

O--g(Qi).

However, As enters into the definition of pl in (3.3a). It is therefore natural to use this extrafreedom by choosing As such that (q, p) satisfies (3.1d), i.e., p and As are the solution of(3.3d)

)Pl po-h Y-i= i (Qi, Pi)+G(Qi)Ai

O= G(ql)Hp (ql, p)

For constrained Hamiltonian systems this projection onto the manifold (3.1 d) does not destroythe symplecticity property shown in Theorem 3.1 (see also [LS94]). The system (3.3d) definesimplicitly p and As in a unique way.

376 LAURENT JAY

Several definitions of the numerical Lagrange multiplier ,kl are conceivable. One possi-bility is to define .1 such that (ql, Pl, 1) satisfies (3.1e), i.e., )1 is given by

(3.3e) )1 (aHpTpaT) -1 (aq(gTp HTp )nt-aHTpqHTp --aHTppH) (ql, Pl).

However, a very accurate value for/.1 may be unnecessary. This remark is important if onewants to avoid the computation of extra derivatives such as Gq. For PRK methods whichsatisfy Cs =’s 1 a fairly good choice is often given by ;kl :-- As.

Because of the singularity of the matrix A due to (3.10), the nonlinear system (3.11)does not possess a solution in general. This remark applies specifically to the cases where thecoefficients (A, b, c) are those of the Radau IIA and Lobatto IIIC methods, and (A, b, c) thoseof their dual symplectic methods. As it seems obviously necessary to have as many unknownsas equations in (3.11), the only supplementary reasonable assumption to make on the PRKcoefficients is

(3.12) aaj--O for j-1 s.

Thus we get Q-q0, and g(Q)--g(qo)-O is automatically satisfied if q0 is consistent. Forsymplectic PRK methods (see (2.3)) satisfying bi 70, this assumption implies that

(3.13) ajl bi for j s

and conversely. Under the assumptions (3.10) and (3.12) the local existence and2niquenessof the PRK solution (qa, Pl, )1) can be shown provided h is sufficiently small, bs : 0, andAoAo is invertible where

(3.14) Ao ".. Aoasl ass asl as,s-1

In this situation, for an efficient solution of the nonlinear system (3.11) (the unknowns areQ2 Qs, P1 Ps, A1 As_l), simplified Newton iterations with the approximateJacobian matrix

l(s-1)n -hAo (R) Hrpp(qo, Po) 0

0 Is -h"o (R) Gr (qo)

Is-1 @ G(qo) 0 0

lead to very simple iterations (see also [HLR89, 7]). Only the decomposition of the matrix

(GHTpp GT)(qo, Po) is needed, and at each iteration s-1 independent linear systems of dimen-sion rn must be solved, this remark being important for a parallel implementation. Due tothe invertibility of AoAo and of (GHpTpGT)(qo, Po), the matrix given in (3.15) is invertible.Another important remark is that high order methods allow the use of larger stepsizes than loworder methods. However, the higher the order of the method, the more Newton-type iterationsare necessary to preserve this order, and the larger the number s of involved internal stages isrequired. Therefore a trade-off between a high order and a low number s of internal stagesmust be made for an efficient implementation.

We mention that a particular class of PRK methods is given by the half-explicit methods(HEMs) of Ostermann (see [O93]), whose coefficients satisfy

(3.16)ajl alj a2j O, asj bj a_,j bj for j s,

aij =a"]j =0 if i<_j for i-1 s, j--1 s.

PRK METHODS FOR CONSTRAINED HAMILTONIAN SYSTEMS 377

However, no second order HEM constructed in [093] is symplectic. The exception is thefirst order HEM already presented in [090] and [HLR89, p. 90], which is symplectic if theHamiltonian is separable.

4. Convergence results. We first introduce the following additional symplifying as-sumptions which enter into the analysis of PRK methods:

c(a) aijjlC-2 Ckij--1 1--1 k(k- 1)

D(R) bicki_2aijj bl blli=1 j=l

k k-1

for i=1 s, k=2 Q,

k-2 R.

If the RK coefficients (A, b, c) satisfy C(q) and D(r), and (A, b, c) satisfy C(’) and D(7")then it can be easily shown that Q > min(q, ’+ 1) and R > min(7", r + 1).

From the discussion of the preceding section we only consider here the class of PRKmethods with coefficients (A, b, c)-(A, b,) satisfying the following hypotheses:

alj 0, asj bj, ajs O, bj bj, cj cj for j s,

H AoAo is invertible, bs 0,

B(p), C(q), O(r), (), (, C(Q), D(R).A detailed analysis of the application of such methods to semi-explicit index-3 DAEs inHessenberg form is given in [J94]. Here we only state the main results of [J94] related toHamiltonian systems with holonomic constraints. The analysis of the application of colloca-tion and RK methods to semi-explicit index-3 DAEs in Hessenberg form is given in [HLR89,6], [J93b], [HJ93], [J94], and [J95]. For PRK methods, slight modifications with regard toRK methods are necessary.

THEOREM 4.1 (local error). (a) Consider the Hamiltonian system with holonomic con-straints (3.1), consistent initial values (qo, Po, )o) at to, and the PRK method (3.3) with

coefficients (A,b,c)-(, ,") satisfying the hypotheses given by H. Then we have

(4.1) qh(tO) (Q(hk+l) 6ph(tO) O(he+l) )h(tO) (Q(he+l)

where

(4.2)

k min(p, 2q +2, 2’+2, q +r+ 1, q +7"+2, ’+r +2, ’+7"+ 1, 2 Q, Q +7, Q+R),

min(p, 2q +2, 2’+2, q +r + 1, q +7"+ 2, ’+r +2, ’+’+ 1, 2Q 1, Q +7, Q+R).

(b) If in addition the hypotheses ofTheorem 2.2 are satisfied then we obtain

k min(p, 2q+2,2r+2, q+r+l,2Q, Q+q),(4.2’)

min(p, 2q+2,2r+2, q+r+l,2Q-1).

The proof of part (a) which is given in [J94] is rather technical. It makes use of a"rooted-tree-type" theory about the Taylor expansion of the exact and the numerical solutions.The most difficult part is to estimate the local error of the p-component. Part (b) is a directconsequence of part (a) since ’-r and7"= q (see the first remark after Theorem 2.2). [3

bc for /=1 s,k(k- 1)

378 LAURENT JAY

THEOREM 4.2 (global error). Under the same hypotheses stated in Theorem 4.1 we have

for tn to nh < Const

(4.3) qn-q(tn) O(he) pn-p(t,) O(he) )-.(tn) (9(he),

where g is the value given in (4.2)-(4.2’).Remark. This theorem remains valid in the case of variable stepsizes with h max/hi.The proof which is given in [J94] is similar to those of [HLR89, Thm. 6.4] and [J95,

Thm. 6.1 ]. It makes use of the following recursion between two neighboring PRK solutions,

(Pq)n+l Aqn+l (Pq)nAqn+ 0 (hll(Pq)nAqnll+hll(Pp)nApnl])(4.4)

(Pp)n+l Apn+l (Pp)nApn+ 0 (hll(Pq)Aqll+hll(Pp)Apll)where Pq and Pp are projectors defined by

S := Gr(GHpGr)-IG(4.5)

Qq "= HpTpS, Pq "= i-eq Qp .= SHp, Pp .= I-Qp.

The couples of s-stage Lobatto IIIA-IIIB methods (see 2) satisfy the simplifying as-

sumptions C(s), D(s-2, C(s-2),2nd D(s). Hence CC(s-1) and DD(s-1) must hold. Infact they also satisfy CC(s) and DD(s), and this is the subject of the following lemma.

LEMMA 4.3. Suppose that alj 0 and asj bj hold for j s, and that the

hypotheses of Theorem 2.2 are fulfilled with p 2s 2, q s 1, and r s 2. Then CC (s)and DD(s) aresatisfied.

ProofofCC(s). Because of Q > s 1 it is sufficient to show that the coefficients

s-2 Ci(4.6) 3i :-- aijajkck= = s(s- 1)

vanish for 1 s. From aaj =0 and Cl =0 we have 3 =0. Using asj =bj and c- we

get

3s bjc-2- ( b(l_c)c_2_s(s-1) s(s-1)

(4.7) =1 j= =n(s 1 1

s-1 s s(s-1)

We will next show that the sums

(4.8) Sm "= bic-l ii=1

vanish for m s- 2. This will give the desired result i 0 for 2 s- 1. Bythe symplecticity condition aij =bj(1 -i we get Sm= Am Bm -Cm where

(4.9)i--1 k=l j=l

(4.10) nm -bj m-1 -" CSk-2ajic ajkj=l i=1 k=l

PRK METHODS FOR CONSTRAINED HAMILTONIAN SYSTEMS 379

(4. ) Cm

_bic.+-s(s- 1) i=l

Each term can be computed separately:

(4.12) Am () bicrf-1 --bk(1--Ck)CS-2 B(--s) (i=l k=l

rn s--1 s ms(s 1)

(4.13) Bm b(l_cr+l)cS_2

m(m+ 1) =1

B(2s___-2) {m(m+l) s-1 m+s m(m+-s)(s-1)

(4.14) Cm B(2s--2)(m +s)s (s 1)

From these results we easily get Sm --O.Proof of DD(s). Because of R > s it is sufficient to show that the coefficients

( s)c ck(4.15) /z := bic-2aijjk--bk/--1 j--1

S S-- S(S-- 1)

vanish for k s. From js =0 (see (3.7)) and cs we have/zs =0. Using jl bl(see (3.10)) and Cl =0 we get

(4.16) ]o61 bl bi c 0i=1

S

We will next show that the sums

(4.17) rn--1

k=l

vanish for m- s- 2. This will give the desired result # -0 for k 2 s- 1. Bythe symplecticity condition jk =b(1 -aj /bj) we get Tm Om Em Fm where

(4.18)i=1 j--1 k=l

(4.19) m--1s--2 aijbkCkEm bi c -j akj

i=1 j=l k=l

(4.20) Fm= bkckS

k=ls-

=1bkck +" t.. ..m+s-

s(s- 1) k--1tJkCk

380 LAURENT JAY

Each term can be computed separately"

(4.21)m-1 B(s) 1

Dm c(1)bic bkCk ms

i=1 k=l

(4.22) Em D(s-2)bici aij c)m

i=1 j--1

s-lC(s-1)bicim

i=1

_m+s-1bicire(m+ 1) i=1

B(2s-2) 1

ms m(m+l)(m+s)

(4.23) Fm B(2s2)ms (m+l)(s 1)

+s(s-1)(m+s)

From these results we easily get Tm O. [3

From Theorem 4.1, Theorem 4.2, and Lemma 4.3 we have the following convergenceresult.

COROLLARY 4.4. For the symplectic couples of s-stage Lobatto IIIA-IIIB methods ap-plied to the constrained Hamiltonian system (3.1) (see (3.3)) with consistent initial values(qo, Po, ,ko) at to, the global error satisfies, for tn-to-nh < Const,

(4.24) qn-q(tn) O(hzs-2) pn-p(tn) O(h2S-2) )n-,(tn) O(h2S-2)

Proof. The symplecticity of Lobatto IIIA-IIIB methods has been proved in[Sun92]. These methods satisfy the hypotheses given by Theorem 2.2 and H with p 2s-2,q =7"=s, r =’--s-2, and Q-R-s. The invertibility of the matrix A00 simply followsfrom CC(s). ]

For separable Hamiltonian systems the 2-stage Lobatto IIIA-IIIB method is half-explicitand is equivalent to the Rattle algorithm proposed in [An83] (see also [LS94]).

Because of the presence of the "explicit" stage Ps in (2.1) and (3.3) for symplectic PRKmethods satisfying asj --bj, the Lobatto IIIA-IIIB methods are not appropriate when solvingstiff Hamiltonian systems, e.g., Hamiltonian systems containing a strong potential of the form

(4.25)e2V (q), 0 < e << 1

This has been numerically observed when trying to solve the stiff spring pendulum equations(see [Lu93] and [HLR89, pp. 10-12]) with Lobatto IIIA-IIIB methods.

For the long-time integration of Hamiltonian systems, a constant-stepsize application ofsymplectic methods performs generally better than variable-stepsize algorithms if the time-scale does not vary greatly along the solution (see [CSS92]). The reason lies in a "backwardanalysis" argument (see [SS92] and [H94]). For constant stepsizes and symplectic methods, thenumerical solution can be interpreted as the exact solution of a nearby perturbed Hamiltoniansystem. It is likely that this result can be extended to Hamiltonian systems with holonomicconstraints. We also point out that the construction of an embedded PRK scheme is not crucialfor a constant-stepsize implementation if an approximation to the error of the method is notneeded.

PRK METHODS FOR CONSTRAINED HAMILTONIAN SYSTEMS 381

FIG. 5.1. The phase portraits (x, px) and (z, Pz) of the 3-stage Lobatto IIIA-IIIB method applied to Examplewith stepsize h O. 12.

5. Numerical experiments. We first notice that for the solution of the nonlinear system(3.11), the s-stage Lobatto IIIA-IIIB method requires a computational work approximatelyequivalent to that arising for the (s-1)-stage RK methods Radau IIA and Gauss. Hence thesemethods are comparable.

Example 1. We consider the motion of a particle of mass m and electric charge e, movingon a sphere of radius R under the action of forces due to an electric field (0, 0, E)r and toa magnetic field (0, 0, B) r (see [Cho92, Problem 7.16]). We use the cartesian coordinatesq (x, y, z) r for the description of the position of the particle. The holonomic constraint isexpressed by

(5.1) 0-- V/x2+y2-+-z2-R

Denoting the generalized momenta by p (px, py, pz), the Hamiltonian of the system isgiven by

1

where co eB/(2mc) and c is the speed of light. This is an example of a nonseparableHamiltonian.

We have applied 5000 steps of the 3-stage Lobatto IIIA-IIIB method of order 4 withstepsize h 0.12,

(5.3) m 1, co 1, R 1, eE 1,

and consistent initial values

(5.4)x(0) 0.2, y(0) 0.2, z(0) .92,

Px (0) 1, py (0) 1, Pz (0) O.

We have plotted in Fig. 5.1 the phase portraits (x, Px) and (z, Pz).In Fig. 5.2 we have drawn the first 500 steps of the numerical Hamiltonian, whose value

for the exact solution is H 1.22- V’-0-.92 0.4808336955. The remaining 4500 steps show

382 LAURENT JAY

0.480840

H

]rie

0.48082510 20 30 40 50 60

FIG. 5.2. The numerical Hamiltonian ofthe 3-stage Lobatto IIIA-IIIB method applied to Example with stepsizeh-0.12.

FIG. 5.3. The phase portraits (x, Px) and (z, Pz) of the 2-stage Radau IIA method applied to Example withstepsize h O. 12.

0.6

H

exct

0,30 ,,,,110 201 ;0 401 501 60

FIG. 5.4. The numerical Hamiltonian ofthe 2-stage Radau IIA method applied to Example with stepsize h =0.12.

the same periodic behavior. If the scale of Fig. 5.4 would be used here, then the numericalHamiltonian would appear nearly equal to the exact value.

As a comparison we have applied the 2-stage Radau IIA method with the same stepsizeh =0.12. The numerical results are given in Figs. 5.3 and 5.4.

Since the numerical solution of the Radau IIA method does not satisfy all underlyingconstraints, we have also applied this method with projections onto these constraints afterevery step (see [AP91], [J94], and [J95] for similar ideas). Although the theoretical orderof convergence is improved compared to the unprojected method, the numerical results did

PRK METHODS FOR CONSTRAINED HAMILTONIAN SYSTEMS 383

not exhibit any visible difference with regards to Figs. 5.3 and 5.4. For that reason thecorresponding figures are not plotted.

We observe that for the Lobatto IIIA-IIIB method the numerical Hamiltonian remainsin tolerable bounds, but it drifts away from the exact value (roughly linearly with time) forthe unprojected and projected Radau IIA methods. This is a demonstration of the differentbehaviors of symplectic and nonsymplectic integrators.

Example 2. The double pendulum. We use the cartesian coordinates q (x l, zl) T,q2 (x2, z2) T for the description of the position of each pendulum. The two holonomicconstraints on the lengths 1 and 2 of the two pendula are

0 V/(X2--Xl)2-qt-(z2-zl)2-e2

Denoting the generalized momenta by pl (Px, Pzl)v and p2 --(Px2, Pz2), the Hamiltonianof the system is given by

(5.6) 2 2H(q, p)2m

(p2xl +Pz)+-m2 (p2x +Pz)+mgz +mzgz:z

and is separable.We have applied 5000 steps of the 3-stage Lobatto IIIA-IIIB method of order 4 with

stepsize h 0.12,

(5.7) m m2, el --2, g 1,

and consistent initial values

(5.8)Xl (0) 0.5, Zl (0) --0v/-.75, X2(0) 0, Z2(0) -2--,

p, (o) o, pz, (o) o, px (o) o, p (o) o.

We have plotted in Fig. 5.5 the phase portraits (xl, Px,), (Z2, Pz2), and in Fig. 5.6 the first 500steps of the numerical Hamiltonian whose value for the exact solution is H -3.75-2.5980762113.

As a comparison we have applied the projected 2-stage Gauss method and the unprojectedand projected 2-stage Radau IIA methods to this problem with the same stepsize h --0.12.Their numerical Hamiltonian is plotted in Fig. 5.7. We point out that the unprojected 2-stage Gauss method generally diverges when applied to Hamiltonian systems with holonomicconstraints (see [HJ93]). This has been numerically observed for this problem. Althoughthe unprojected Gauss methods are symplectic, the projected Gauss methods are not, and weclearly see here that the numerical Hamiltonian drifts off the exact value. However, this driftis less drastic here than for the unprojected Radau IIA method, which in turn is less severethan for the projected Radau IIA method.

In Fig. 5.8, as a last experiment, the global errors at 5 of the four above-mentionedmethods have been plotted as functions of h. Since we have used logarithmic scales, thecurves appear as straight lines of slope k whenever the leading term of the error is O(h). Thisbehavior is indicated in the figures.

The order of convergence of the projected s-stage Gauss method is s and that of theprojected s-stage Radau IIA method is 2s- 1 (see [J93b], [J94], and [J95]). For the unpro-jected s-stage Radau IIA method the order of convergence is 2s- for the q-component, s forthe p-component, and s-1 for the ,k-component. The predicted orders are confirmed in Fig. 5.8,

384 LAURENT JAY

FIG. 5.5. Thephaseportraits (Xl, Px and (Z2, Pz2 ofthe 3-stage Lobatto IIIA-IIIB method applied to Example2 with stepsize h =0.12.

-2.598066

H

J/exact

-2"5980860-’ 10 20 ;0 iO 50 60

FIG. 5.6. The numerical Hamiltonian ofthe 3-stage Lobatto IIIA-IIIB method applied to Example 2 with stepsizeh--0.12.

-2.58

H

/’/exact

Radau IIA (s 2)

_2.640 10 20 30 40 50 60

FIG. 5.7. The numerical Hamiltonians of the projected 2-stage Gauss method and of the unprojected andprojected 2-stage Radau IIA methods applied to Example 2 with stepsize h =0.12.

and this clearly shows the superiority of the Lobatto IIIA-IIIB schemes, also in terms ofaccuracy.

Acknowledgments. I am grateful to E. Hairer for his remarks during the preparation ofthis paper. I express my gratitude to G. Wanner for his comments on a preliminary version ofthis article. I wish to thank S. F. Bernatchez for her careful reading of the English part of themanuscript. Special thanks go to the unknown referees for their constructive criticisms.

PRK METHODS FOR CONSTRAINED HAMILTONIAN SYSTEMS 385

Ilq,-q(t,)li IlPn-p(t,)l[

h

h.h

h4

l10-3 10-2 10-1

10-5

10-to

h lO-3 lO-2 lO-1

lO-5

hz

h3lO-tO

h4

H(q,, p,)- H(q(tn ), p(tn ))1

10-5

10-1oh2

h4

FIG. 5.8. Global errors at 5 offour methods applied to Example 2 (projected 2-stage Gauss: " projected2-stage Radau IIA: ; 2-stage Radau IIA: -4-; 3-stage Lobatto IIIA-IIIB" 0).

[ASS931

[An83]

[Ar89]

[A85]

[AMR88]

lAP911

[AP92]

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