higher order symplectic methods based on the modi ed

M a t h e m a t i c aB a l k a n i c a

—————————NewSeries Vol. 26, 2012, Fasc. 3–4

Higher Order Symplectic Methods Based onthe Modified Vector Fields

Gamze Tanoglu 1, Hakan Gunduz 2

Presented at MASSEE International Conference on Mathematics MICOM-2009

The higher-order, structure-preserving numerical integrators based on the modified

vector fields (backward error analysis ) are used to construct discretizations of separable sys-

tems. This new approach is called as modifying integrators. As an example of modifying

integrator Lobatto III-A and III-B are chosen. These new efficient higher-order numerical

integrators are applied to the separable Hamiltonian equations.

MSC 2010: 65L06.Key Words: structure-preserving numerical integrators, Lobatto III-A

and III-B integrator, separable Hamiltonian equations.

1. Introduction

In recent years, geometric numerical integration methods have come [1]to the fore, partly as an alternative to traditional methods such as Runge-Kuttamethods. A numerical method is called geometric integrator if it preserves oneor more physical/geometric properties of the system exactly (i.e up to round-off error). Examples of such geometric properties [4] that can be preservedare (first) integrals, symplectic structure, symmetries and reversing-symmetries,phase-space volume, Lyapunov functions, foliations, e.t.c.

In the literature symplectic methods are generally constructed usinggenerating functions, Runge-Kutta methods, splitting methods and variationalmethods. One of the methods for constructing high-order symplectic integra-tors is developed by using the idea of backward error analysis while constructingmodified equations by inverting the roles of the exact and numerical flows. Theidea given in [1] is presented as follows: We consider an initial value problem

y = f(y), y(0) = y0 (1)

430 G. Tanoglu, H. Gunduz

where f ∈ C∞[Rn] and a numerical one-step integrator yn+1 = Φf,h(yn). Wesearch for a modified differential equation

z = fh(z) = f(z) + hf2(z) + h2f3(z) + ..., z(0) = y0 (2)

such that the numerical solution {zn} of the method applied with step size hto (2) yields formally the exact solution of the original differential equation (1),i.e. zn = y(nh) for n = 0, 1, 2, ... The coefficient functions fj(z) can be computedrecursively.

Having found first functions fj(z), one can use the truncation

z = f[r]h (z) = f(z) + hf2(z) + ... + hr−1fr(z) (3)

of the modified differential equation corresponding to Φf,h(y). A numericalmethod zn+1 = Φ

f[r]h ,h

(zn) approximates the solution of (1) with order r. It was

called a modifying integrator because it applies to the modified vector field f[r]h

instead of f(y). In this study we mainly consider the partitioned and separablesystems given in (4) and (5), {

y′ = f(y, z)z′ = g(y, z)

(4)

{y′ = f(z)z′ = g(y).

(5)

In particular, we will be interested in canonical Hamiltonian equations,which are generated by a Hamiltonian function H(y, z):

f(y, z) = Hz(y, z), g(y, z) = −Hy(y, z) (6)

Such system often arise in mechanical systems described by a Hamilto-nian function

H =p2

2+ V (q),

which provides us with the Hamiltonian equations of motion{q′ = pp′ = −Vq(q)

(7)

In next section we construct the higher order method corresponding to LobattoIIIA-IIIB method by using the modified vector fields.

Higher Order Symplectic Methods ... 431

2. Construct of Higher Order Modified Lobatto IIIA – IIIBpair

In this section we will construct higher order method corresponding toLobatto IIIA-IIIB pair of method. Our Lobatto IIIA - IIIB pair of order 2 isgiven as follows:

k1 = f

(y0, z0 +

h

2l1

), k2 = f

(y0 +

h

2(k1 + k2), z0 +

h

2l1

),

l1 = g

(y0, z0 +

h

2l1

), l2 = g

(y0 +

h

2(k1 + k2), z0 +

h

2l1

),

y1 = y0 +h

2(k1 + k2), z1 = z0 +

h

2(l1 + l2).

In the case of separable equations this method gets reduced to an explicitform known as Stormer–Verlet scheme.

The modified differential equations are{y′ = f(y, z) + h2a(y, z),z′ = g(y, z) + h2b(y, z),

(8)

where

a(y, z) =1

12

(−fyy(f, f) + fyz(f, g) +

1

2fzz(g, g)− fyfyf−

− fyfzg + 2fzgyf − fzgzg)

b(y, z) =1

12

(−gyy(f, f) + gyz(f, g) +

1

2gzz(g, g)− gyfyf−

− gyfzg + 2gzgyf − gzgzg)

If the original equations are Hamiltonian, then the modified differentialequations (8) are generated by the Hamiltonian function

H [3] = H +h2

12

(−Hyy(f, f) + Hyz(f, g) +

1

2Hzz(g, g)

),

where f and g are given by (6).The modified differential equations for separable systems are simplified:


a(y, z) =1

12

(1

2fzz(g, g) + 2fzgyf

),

b(y, z) = − 1

12(gyy(f, f) + gyfzg).

If the original equations are Hamiltonian, the modified differential equa-tions are generated by

H [3] = H +h2

12

(−Hyy(f, f) +

1

2Hzz(g, g)

)For mechanical system, equation (7) can be

H [3] = H +h2

12

(−Vqq(p, p) +

1

2(Vq, Vq)

)The modified equations take the form

q′ = p +h2

6gqp =

(I − h2

6Vqq

)p

p′ = −Vq(q)− h2

12(gqq(p, p) + gqg)

Application of Lobatto IIIA - IIIB pair of order 2 to these equations leadsto a scheme which can be split into three stages:

p1/2 = p0 +h

2

(g(q0)−

h2

12

(gqq(q0)(p1/2, p1/2) + gq(q0)g(q0)

)),

q1 = q0 + h

(I +

h2

12(g(q0) + g(q1))

)p1/2,

p1 = p1/2 +h

2

(g(q1)−

h2

12

(gqq(q1)(p1/2, p1/2) + gq(q1)g(q1)

)),

where we introduced an intermediate variable

p1/2 = p0 +h

2l1.

Note that the first and second stages are implicit and the third is explicit.


3. Order of the Modified Lobatto Method

Proposition 3.1. Application of the Lobatto IIIA - IIIB pair of thesecond order to the modified differential equations gives a numerical method oforder 4.

P r o o f. Consider the mechanical system as we mentioned before givenin the equations (5),

q = p

p = −Vq(q) (9)

Applying the hamiltonian equation to mechanical system givesq′ = p +

h2

6gqp =

(I − h2

6Vqq

)p

p′ = −Vq(q)− h2

12(gqq(p, p) + gqg)

(10)

Since Lobatto IIIA - IIIB pair of order 2 is given as follows:

k1 = f

(q0, p0 +

h

2l1

), k2 = f

(q0 +

h

2(k1 + k2), p0 +

h

2l1

),

l1 = g

(q0, p0 +

h

2l1

), l2 = g

(q0 +

h

2(k1 + k2), p0 +

h

2l1

),

qn+1 = qn +h

2(k1 + k2), pn+1 = pn +

h

2(l1 + l2).

The Taylor expansions of k1, k2, l1, l2 are:

k1 = f(q0, p0) +h

2l1∂f

∂p(q0, p0) +

h2

2!l21∂2f

∂p2(q0, p0) + ... (11)

= p0 −h2

6gqp0 + hl1 −

h3

6l1gq

We need to find l1, since we use l1 in k1;

l1 = −Vq(q0)−h2

12(gqq(p0, p0)+gqg)+

h

2l1[−Vqq(q0)−

h2

12(gqqq(p0, p0)+gqqg+g2q )]+...

q1 = q0 + h2 (k1 + k2) and since k1 = k2, then

q1 = q0+h

2(k1+k2) = q0+hp0−

h3

6gqp0+h2(−Vq(q0)−

h2

12gqq(p0, p0)−

h2

12gqg+...)−


−h4

6(−Vq(q0)−

h2

12gqq(p0, p0)−

h2

12gqg + ...)gq

Our exact solution of q = p + h2

6 gqp is

q(t0 + h) = q(t0) + hq(t0) +h2

2!q(t0) +

h3

3!q(3)(t0) +

h4

4!q(4)(t0) + ...

Substituting q = p + h2

6 gqp in taylor series expansion, we get

q(t0 + h) = q0 + h(p0 −h2

6gqp0) +

h2

2!(−Vq(q0)−

h2

6gqq −

h2

6gqg) (12)

+h3

3!(−h2

6gqqq − ...) +

h4

4!(Vq(q0) +

h2

3gqqgq +

h2

3g2qg) + ...

Substracting q1 from q(t0 +h), we get h3

3! (−h2

6 gqqq − ...) = C1h5 completes first

part of proof.For the second part of proof, we need l1.

l1 = g(q0, p0 +h

2l1) = g(q0, p0) +

h

2l1gq +

h2

4l21gqq + ...

l1 =− Vq(q0)−h2

12(gqq(p0, p0) + gqg) +

h

2l1[−Vqq(q0)−

h2

12(gqqq(p0, p0)+

+ gqqg + g2q )] + ... ∼= g

⇓

l1 =− Vq(q0)−h2

12gqq(p0, p0)−

h2

12gqg −

h

2gVqq(q0)−

h3

24gqqqg −

h3

24gqqg

2−

− h3

24g2qg +

h2

4g2gqq + ...

and

l2 = g(q0 +h

2(k1 + k2), p0 +

h

2l1) =

= −Vq(q0+hp0−h3

6gqp0+h2g−h4

6gqg)−h2

12(gqq(p0, p0)+gqg)+

h

2ggq+

h2

4g2gqq+...

Since

p1 = p0 +h

2(l1 + l2) =

= p0−hVq(q0)−h3

12gqq(p0, p0)−

h3

12gqg −

h2

2Vqq(q0)−

h4

24gqqq(p0, p0)−

− h4

24gqqg −

h4

24g2q −

h3

6Vqqq + ...


Our exact solution of p = −Vq(q)− h2

12 (gqq(p, p) + gqg) is

p(t0 + h) = p0 + h(−Vq(q0)−h2

12gqq(p0, p0)−

h2

12gqg) (13)

+h2

2!(−Vqq(q0)−

h2

12gqqq(p0, p0)−

h2

12gqqg −

h2

12g2q )

+h3

3!(−Vqqq(q0, q0)−

h2

12gqqqq −

h2

12gqqqg −

h2

12gqqgq − ...) +

h4

4!...

Substracting p1 from p(t0 + h), we get

h3

3!(−h2

12gqqqq −

h2

12gqqqg −

h2

12gqqgq − ...) = C2h

5

This completes the second part of the proof.As can be seen that, we find errors

q(t0 + h)− q1 = C1h5 (14)

p(t0 + h)− p1 = C2h5

So our new modified Lobatto method is a method of order 4.

4. Numerical Test Problem

4.1 Applications to Harmonic Oscillator System

We determine the modified differential equations based on the LobattoIIIA–IIIB method for the linear Hamiltonian system, as an example of HarmonicOscillation system. We then illustrate the trajectory of motion (phase space),energy error ‖H(q, p) − H(q0, p0)‖2 and global error ‖yexact − ynumeric‖2 wherey = y(q, p) . The Hamiltonian of this system can be given as

H(q, p) =1

2p2 +

1

2q2 (15)

so that the equations of motion become

q = Hp(q, p) = p (16)

p = Hq(q, p) = −q (17)


4.2 Modified Equations Based on Lobatto IIIA-IIIB Method

Harmonic oscillation problem can be read as

q = p = f(p) (18)

p = −q = −Vq(q) = g(q). (19)

In this section we derive the modified differential equations of the system (18)and (19) based on Lobatto IIIA–IIIB methods. Applying (18) and (19) to themodified equations

q′ = p +h2

6gqp =

(I − h2

6Vqq

)p

p′ = −Vq(q)− h2

12(gqq(p, p) + gqg)

we obtain the modified differential equations of the system (18) and (19) in theform

q′ =

(1− h2

6

)p (20)

p′ = −(

1 +h2

12

)q (21)

4.3 Numerical Implementation for Harmonic Oscillation

In this section we apply Lobatto IIIA–IIIB pair of order 2, LobattoIIIA–IIIB pair of order 4, Runge Kutta method(order 4), Modified Midpointmethod(order 4), Gauss Collocation method(order 4) to the system (18) and(19) and Modified Lobatto method of order 4 to the system (20) and (21) in or-der to show effectiveness. Comparison of these methods for pinitial = 1; qinitial =0;h = .01; t = 0 : h : 100;n = 100

h+1 are illustrated in Figure 1 and 2 with respectto the energy error, global error. We observe that since all of these methods aresymplectic, the shape of the trajectory preserved. Comparison of norm of globalerrors, norm of energy errors and CPU times for the methods we consider areillustrated in the Table 1.


Figure 1: Energy error and global error in Hamiltonian by 2-nd order Lobattomethod, 4-th order Lobatto method and 4-th order Runge-Kutta method (fromleft to right).

4.4 Comparison of the norm of global error, norm of energyerror and CPU time

METHODS I II III

Lobatto method of order 4 0.0225 2.5342e-004 1.085Runge Kutta method of order 4 3.4242e-007 4.0099e-009 0.577

Modified Midpoint of order 4 3.4245e-007 2.2983e-011 0.733Gauss Colloc. method of order 4 0.7056 1.6687e-013 2.366Modi. Lobatto method of order 4 3.3803e-007 1.6428e-013 0.589

Table 1: Comparison of (I) the norm of global errors, (II) norm of energyerrors and (III) CPU times in Hamiltonian

5. Conclusion

In the paper, we constructed 4-th order Modified Lobatto IIIA-IIIB pairmethod by using modified vector field idea. One can develop higher ordermethod in the same manner. We showed that the consistency of the methodand order of the method. From the simulation based on the harmonic oscillationproblem the following results are revealed:


Figure 2: Energy error and global error in Hamiltonian by 4-th order ModifiedMidpoint method, Gauss Collocation method of order 4 and 4-th order ModifiedLobatto method (from left to right).

• As the error of the methods we consider are increasing, the energy errorsand global errors in solutions are decreasing.

• Comparising of the 4th order methods showed that modified lobatto methodis more efficient with respect to the energy errors, global errors and CPUtimes.

In future we will apply our method to nonlinear ODE and PDE problems.

6. Acknowledgement

The results presented in this paper partially obtained during the visit ofRoman Kozlov to Izmir Inst. of Tech. in July, 2008. His visit was supported bythe Scientific and Techonological Researh Council of Turkey (TUBITAK).

References

[1] E. H a i r e r, C. L u b i c h, G. W a n n e r. Geometric Numerical Integration.Structure-preserving algorithms for ordinary differential equations. SpringerSeries in Computational Mathematics, Vol. 31, Springer-Verlag, Berlin,2002.

[2] P. C h a r t i e r, E. H a i r e r, G. V i l m a r t. ESAIM: Proceedings, 2006, 16–20.[3] K. S t r e h m e l, R. W e i n e r. Partitioned Adaptive Runge-Kutta Methods

and Their Stability. Springer-Verlag, 1984, 283–300.


[4] C. J. B u d d, M. D. P i g g o t t. Geometric Integration and Its Applications// Handbook of Numerical Analysis, Vol XI, 2000, 35–139.

Institute of Technology Received 04.02.2010Izmir Institute of Technology Gulbahce KampusIzmir, 35437, TURKEY1 E-MAIL: [email protected] E-MAIL: [email protected],

higher order symplectic methods based on the modi ed

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