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29 Sep 1993

Comparative Study of Louville and Symplectic Integrators Comparative Study of Louville and Symplectic Integrators

Daniel I. Okunbor

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Recommended Citation Recommended Citation Okunbor, Daniel I., "Comparative Study of Louville and Symplectic Integrators" (1993). Computer Science Technical Reports. 47. https://scholarsmine.mst.edu/comsci_techreports/47

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Comparative Study of Louville and

Symplectic Integrators 1

Daniel I Okunbor

CSC-93-26

September 29, !993

Department of Computer Science University of Missouri-Rolla

Rolla, Missouri 65401

1This paper has been submitted to J. Comp. & Appl. Math, for publication. This work was supported in part by NSF Grant DMS 90 15533 while the author was at the University of Illinois at Champaign-Urbana

Comparative Study of Louville and SymplecticIntegrators

Daniel I. Okunbor Department of Computer Science

University of Missouri-Rolla Rolla, Missouri 65401

September 29, 1993

AbstractIn this paper, we construct an integrator that conserves volume in phase space. We com­

pare the results obtained using this method and a symplectic integrator. The results of our experiments do not reveal any superiority of the symplectic over strictly volume-preserving integrators. We also investigate the effect of numerically conserving energy in a numerical process by rescaling velocities to keep energy constant at every step. Our results for Henon- Heiles problem show that keeping energy constant in this way destroys ergodicity and forces the solution onto a periodic orbit.

Key Words: Hamiltonian systems, energy conservation, symplectic integrators, Louville inte­grators

1 IntroductionThe problem of interest is that of a computer solution of a Hamiltonian dynamical system of the form

rjy- = JV H (z ,t) , z(t0) = z0, (1)

where z € 'R?1', t is time, H (called the Hamiltonian) is a scalar function, is the skew-symmetric

matrix ^ ^ ^ and the two identity matrices I are of equal dimension. We assume that the

function H (z,t) is sufficiently smooth to ensure the existence of a unique solution. The value v is the number of the degrees of freedom of the system. This value may sometimes be large, especially in systems obtained from TV-body motion and spatial discretization of partial differential equations. An important type of Hamiltonian is

H (q ,p )= T (p ) + V(q), (2)

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where z = ^ ^ ^ . The functions T(p) and V(q) are associated, respectively, with kinetic energy and

potential energy of the dynamical system. (2) is called a separable Hamiltonian system. Another important type of Hamiltonian system is a subclass of (2) which has the Hamiltonian:

H(q,p) = ±pTM - 1p+ V (q), (3)

where Mis a diagonal matrix with positive diagonal elements. We call (3) a special separable Hamiltonian.

A practical question that readily comes to mind is: given a dynamical system ^ = r(z), how do we determine whether or not this is a Hamiltonian system. To do this, we find a continuously differentiable function H such that T(z) = JV H . If such H exists, then the dynamical system is Hamiltonian. Note that for Hamiltonian system, we can show that the field r(z) is divergence free. That is,

divr(^) = V • T(z) = 0. (4)

For systems with the number of degrees of freedom higher than one, (4) is not a sufficient condition for r(2 ,t) = JV H , we need J -1 Fzto be symmetric. For instance, the geodesic flow problem

^ = |(1 - Qi ~ <ll?Pu= 9i(1 - 9i )>

= \ ( l ~9l ~ 92)2P2, = 92(1 -

Since J -1TZ is symmetric, the system is Hamiltonian with

H(q,p) = ^(1 - 9i - q lf{p \ +p\).

Hamiltonian systems have qualitative features that are very important when they are being integrated. Most of the conventional numerical integrators such as the classical 4-stage, 4th-order Runge-Kutta method do not capture these qualitative features of the systems. Seemingly, all the features exhibited by the flow of the Hamiltonian system are consequences of just one property, namely, the property that the flow of the system is symplectic. The flow is the mapping from a set of initial values to a set of solution values at some time later. In differential-geometry, the solution of the system would be said to have symplectic structure.

The construction of symplectic integrators for Hamiltonian has been the interest of several researchers. Ruth[16] and Feng[5] were the first, independently, to give published reports on the possibility of symplectic numerical integration of Hamiltonian systems. Ruth[16] discovered 1-, 2- and 3-stage methods of orders < 3. Ruth’s work was followed by a considerable research in the area of constructing higher order symplectic integrators[4, 5, 6, 15, 17, 20, 21]. Forest and Ruth[6] derived a symmetric explicit 3-stage symplectic integrator of order 4. Yoshida[21] was the first to indicate the existence of symplectic integrators of arbitrary higher order. He proved the possibility of constructing 3fc-stage method having order 2k + 2 using a composition of symplectic 1-stage method of order 2. He derived numerically 7- and 15-stage symplectic integrators, respectively, of

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orders 6 and 8 using a Lie group approach. Using the discrete variable approach, several Runge- Kutta-Nystrom methods of orders at most 8, some of which are equivalent to the Yoshida’s methods have been constructed by Okunbor and Skeel[14].

This paper is not about the derivation of symplectic integrators, detailed treatment of this are found in [15] and the references therein. We focus on the analysis of existing symplectic integrators. The analyses presented in this paper are different from what are avialable in the litereture [2], So far, the emphasis has been on the comparison between symplectic integrators and non-symplectic integrators[15, 17] and the effect of variable stepsize implementation of symplectic integrators[2]. There are basically two issues that will be presented in this paper. The first is that of energy conser­vation. Can the qualitative behavior of symplectic integrators be explained by linearized stability or simply by energy conservation? As reported by Ge and Marsden[7], symplectic integrators do not conserve energy. However, the property of being symplectic surpasses energy conservation. In section 3, we investigate the effect of numerically conserving energy. To do this, we rescale velocities to keep energy constant at every step. Our results for Henon-Heiles problem show that keeping energy constant in this way destroys ergodicity and forces the solution onto a periodic orbit.

The second issue is that of volume preservation. It is well-known that methods that are sym­plectic preserve volume in phase space. The construction of integrators that merely preserve volume in phase space was considered by Suris[19]. This poses the question as to whether or not volume preservation is all that a numerical integrator requires to represent the qualitative behavior of the flow of the system. We examine this question in somewhat incomplete manner in Section 4. We apply a merely volume-preserving and a symplectic integrator to a two degrees-of-freedom Kepler problem and 16 degrees-of-freedom pseudospectral discretization of a sine-Gordon equation. The results of these experiments do not reveal any superiority of the symplectic over volume-preserving integrator.

2 N um erical IntegrationsA good starting point for the derivation of integrators would be to look at the time flow (h-flow) of a system, where h is the timestep. That is,

/ q((n + l)h)V P((n + W

= $h ,n = 0 ,1 ,...

Hence, 4^ is a mapping of phase space to phase space. The family {$/,} has an identity 4>o and each member 4>/, has an inverse $-h and is differentiable. Also, composition of several $ is possible:

$ t+„ =

where t and u represent two different times. The family {$/,} is called one-parameter group of diffeomorphisms since it consists of one-to-one differentiable functions. As men tioned in the last section, is symplectic. To determine a mapping $ which numerically approximates $ up to certain order of accuracy p and possesses some qualitative characteristics of 4> constitutes a research problem. One wants to find a mapping 4>/j

/ q((n + l)h)\ P((n + 1

n — 0,1,. (5)

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such that$/, = $ h + 0 ( ^ +1)

and

<^>T ''<£**> = wwhere denotes the Jacobian matrix o f the transformation.

Definition 1 A method symplectic for any h and any Hamiltonian system for which it is ap­plicable if its Jacobian matrix satisfies (6).

The conditions for canonical Runge-Kutta integrators, partitioned Runge-Kutta integrators and Runge-Kutta-Nystrom have been given an extensive consideration, see [3, 18, 1]. From the sym­plectic conditions for RK, it is clear that there are no explicit symplectic RK integrators. All the explicit symplectic Runge-Kutta-Nystrom and explicit symplectic partitioned RK can be cast as

Qo QmQ0 = Qni

for * = 1,2, . . . , s,Qi = Qi-i + h(ci - Cj_i)Qt_i, where c0 = 0,

Qi = Qi-i + hBif(Qi),Qn+l - QsT h(l Cg Qg,

Qn+ 1 = Qs-

where Bi and c,- are the parameters defining the symplectic integrators.

3 Conservation of EnergyThe value of the Hamiltonian H(q,p) of a Hamiltonian system is a conserved quantity for given

initial conditions f ^ , that is, H(q(t),p(t)) = H(q(0),p(0)) for all time , where

the solution of the system. Usually H corresponds to the energy of the system. Several numerical integrators for dynamical systems, not necessarily symplectic, that conserve energy have been pro­posed (see [8, 9, 10, 13]). Sanz-Serna[17] claims that the conservation of energy forces the solution orbits of the Hamiltonian system to be in the 2u — 1 dimensional surface, thereby allowing them to be free within the surface and therefore, it may not be as important as the property of being symplectic.

In fact, it has been proved by Ge and Marsden[7] for Hamiltonian systems having no integrals other than the energy that if a symplectic integrator always conserves energy, then it must agree with the map of the exact Hamiltonian system up to a reparametrization of the time.

A simple-minded way to conserve energy is rescale velocities at every step using the formula

pn = spn, s = ^ (H 0 K(gn))/T(pn),

where T(pn) and V(qn) are, respectively, the kinetic energy and potential energy at time tn and Hq is the initial energy. To find the effect that rescaling velocities might have on a numerical integrator, we

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consider the Henon-Heiles problem with the initial condition (gi, = (0,0.2,0.4483395,0)giving energy 0.117835. We compute the solutions using the symplectic 3-stage 3rd-order RKN method derived above, a G-symmetric 5-stage 4th-order method ( acronym : RO) with the following coefficient was constructed by Okunbor and Skeel[15]

c : B

7_

4?3

25

33

24481

23

33

41

1 oand the non-symplectic 3-stage, 4th-order RKN method taken from [11]:

Q2 = qn + ^hqn + ^h2f(q n),

Q3 = Qn + hqn + ~h2f(Q 2),

qn+ 1 = Qn + hqn + ^ h 2f ( q n ) +

Qn+l = Qn + g M / ( & i ) + f ( Q 2) +

(acronym: RKN4). It can be shown that velocity rescaling destroys the sympletic property of RO. Figure 1 shows the results that we obtain using RKN4 and RO with and without rescaling velocities. From Figure 1 we see that keeping the energy constant for the case of RKN4 destroys the ergodicity and forces the solution onto a periodic orbit. Therefore, to conserve energy numerically does not make the results obtained by non-symplectic to be comparable to that obtained using symplectic integrator. The RO method with velocity rescaling is not better than the RO method without rescaling.

We also examine to what order of accuracy symplectic RKN methods conserve energy. That is, would symplectic RKN method of order say, p conserves energy with order of accuracy higher than p? If we denote by qn and pn, the numerical solution, respectively, of q(tn) and p(tn), then the error in energy is

eH(h) = H(q(tn),p(tn)) - H(qn,pn).For a consistent method, the Taylor expansion of error is

OOeH(h) = '£ ,a ih \

i=1

where are in terms of the elementary differentials and method parameters. If a method conserves H to an order of accuracy p, then a; = 0, for i < p. For p < 3, we discovered that the order conditions in energy are the same as the order conditions of the method. The same may also be true for p > 3 that a method of order p conserves energy to same order of accuracy.

4 Liouville vs. Sym plectic IntegratorsThe property of being symplectic can give rise to many qualitative characteristics of Hamiltonian systems. One of these characteristics is the preservation of volume in phase space. The volume element Uz = dz]_dz2 • • • dz2l/ of 2 is related to the volume element = dz\dz2 • • • of z by

Yh = IdetS'jn,,

5

RKN4 without vel. rescaling RKN4 with vel. rescaling

-0.4 -0.2

ql

RO without vel. rescaling-----1----- , , \ , —r—

r * * \ ■'

0.2 h

0

-0.2 h

-0.4

p .*•

* ■ ■ % «, . . * " * m '*

MO'

0.2 0.4 -0.4 -0.2 0.2 0.4

ql

RO with vel. rescaling

(NO'

-0.4 -0.2 0 0.2 0.4

qi

0.2 0.4

qi

Figure 1: The effect of numerically conserving energy

6

where S is the Jacobian matrix of the transformation. Since STJ S = J, then (lets' = ±1, implying that volume is invariant under a symplectic transformation. This result is important when sampling phase space.

A transformation which conserves volume in phase space is called a Liouville transformation. We say that a consistent integrator is Liouville if it gives rise to a Liouville transformation. Clearly, all symplectic integrators are Liouville, but the converse is not in general true. In this section we examine in more details the usefulness, if any, of the symplectic property over the property of being Liouville. In other words, what additional gain does one achieve from symplectic property. Is Liouville property all that numerical integrator needs to represent qualitatively the behavior of the flow of the system? The motivation for such a consideration arises from the fact that the conditions for a method to be Liouville are less restrictive. It is clear that symplectic RKN methods form a subset of Liouville RKN methods. For example, consider a general explicit 2-stage RKN method. The Jacobian matrix S of this method is

„ _ / I -j- Jv^b[D\ H- It b'yD'y + h^b2(i2\DiD2 h + h?b\C\D\ + -!■y h.H\D\ + I1JJ2U2 + h B2CL21U1U2-l + + + J

where D\ and D2 are the derivatives of f(y ) with respect to y evaluated at + c\hp and q + C2 hp + h2a,2if(q+ cihp), respectively. To obtain necessary conditions to be Liouville, we consider the scalar case, and we get three equations, namely,

61 - +62 — B 2 + B2 C2

(62 - B2 + 5 2c i)o2i + (B2bi - Bib2)(c2 - c i)

With equations (7) and (8) satisfied, equation (9) becomes

0, (7)0, (8)0. (9)

( — B 2 CI21 + ^ 2 1 — B\b2)(c2 — Ci) = 0.

To be symplectic, the method must satisfy 21 + B2bi — Bib2 = 0. However, to be Liouville, this is not necessary if C2 = ci. What this means is that if we choose ci so that it is equal to C2, then the method is Liouville but may or may not be symplectic. The method

12 0f? h 0

"T 1 ’i i2 2

is Liouville but not symplectic. One can show that this method is Liouville for systems. This method is of order 2. It has an accuracy comparable to Stormer-Verlet method but requires twice as much work. In what follows, we perform numerical experiments in an attempt to compare symplectic and Liouville integrators. From the numerical results, there is no noticeable difference between a symplectic integrator and the above Liouville method. However, there is a difficulty in constructing methods that are Liouville without assuming symplectic property. The conditions for Liouville property are not explicitly expressible in terms of the method parameters.

In our experiments we compare the above Liouville method and Stormer-Verlet method. Both methods have the same P-stability threshold as indicated in the following section. Therefore, using

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the same timestep for both methods does not pose any serious consequences. We consider three Hamiltonian systems. The first is the vibrating beam problem with

H(q,p) = \{p2 - q 2 ( 1 0 )

and initial conditions (1,0)T. Using a timestep of 0.01, we computed solution for a total of 10000 steps. The plots of the trajectories for both methods are shown in Figure 2 We see from this figure no important difference between the two methods. Next is the 2-body problem with

H{q,p) = \{p\+pl)~~J== ( 1 1 )

and initial condition (0.5,0,0, \/3). The timestep in this case is 0.0001 and total time of the experiment in terms of periods is 3000. The global errors in the trajectory against time for both methods are indicated in Figure 3. Similar to the conclusion reached in the first problem, we notice no significant difference between both methods.

The third problem is the sine-Gordon equation:

utt + uxx + sin 0, (12)u(x,0) = 7r + 0.1 cos (px u<(a;,0) = 0,

where fj, = ^ an(l L — 2\/27r. The solution is periodic in x with period L. This is considered to be a more difficult problem. The paper by Herbst and Ablowitz[12] describes the application of a pseudospectral method to (12). The Hamiltonian of the pseudospectral spatial discretization in Fourier space is

1 I* " 1H = n ! C \PkP-k + pfakq-k] - K$ 3 COsUi ( 1 3 )

k=-\N iV j=-±N

whereiiV-l

Uj = (-F_1)j{?fc} := 5Z qkexpiiPkXj),k=-\N

qN/2 •= q~N/2i Pn/2 '■= P -n/2, Pk = and N is even. The Hamiltonian system is

qk = Pk, Pk = -{p\qk + Fk{sin Uj}), k = - ~ iV , . . . , 1,

where

Fk{vj I ]

We choose N = 16. Figure 4 depicts the solution obtained using a time spacing of 0.02 for a total of 5000 steps for the two methods. Again, there is no noticeable difference between the two methods. On the basis of these three sets of experiments, it may be tempting to say that the reason while symplectic methods perform the way they did might be due to the preservation of volume in phase space. In any case, more needs to be done in this regard.

8

q

q

Figure 2: Vibrating beam problem: Liouville vs. symplectic

9

Figure 3: 2-Body problem: Liouville vs. symplectic

10

Canonical method

L iou ville method

Figure 4: Sine-Gordon equation: Liouville vs. symplectic

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References[1] L. Abia and J. M. Sanz-Serna. Partitioned Runge-Kutta methods for separable Hamiltonian

problems. Technical Report 1990/8, Universidad de Valladolid, Valladolid, Spain, 1990.

[2] M. P. Calvo and J. M. Sanz-Serna. The development of variable-step symplectic integrators, with applications to the two-body problem. SIAM J. Sci. Comput. To appear.

[3] M. P. Calvo and J. M. Sanz-Serna. Order conditions for canonical Runge-Kutta-Nystrom methods. BIT,32:131-142, 1992.

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[5] K. Feng. On difference schemes and symplectic geometry. In K. Feng, editor, Proc. 1984 Beijing Symposium on Differential Geometryy and Differential Equations- Computation of Differential Equations, pages 42-58, Science Press, Beijing, 1985.

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[13] R. A. LaBudde and D. Greenspan. Energy and momentum conserving methods of arbitrary order for the numerical integration of equations of motion II. Motion of a system of particles. Numer. Math., 26:1-16, 1976.

[14] D. Okunbor and R. D. Skeel. Canonical Runge-Kutta-Nystro m methods of orders 5 and 6. J. Comp. & Appl. Math., 1992. To appear.

[15] D. Okunbor and R. D. Skeel. Explicit canonical methods for Hamiltonian systems. Math. Comp., 59(200):439-455, 1992.

[16] R. D. Ruth. A canonical integration technique. IEEE Trans, on Nucl. Sci., NS-30(4):2669- 2671, 1983.

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[17] J. M. Sanz-Serna. The numerical integration of Hamiltonian systems. In J. R. Cash and I. Gladwell, editors, Proc. of IMA Conference on Comput. ODEs. Oxford Univ. Press, 1992.

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