1 Introduction

The oscillator is an ideal model in the field of solid mechanics. Ignoring the damping effect, the vibration process of an oscillator system is a classical conservative system,and the energy in the vibration process is a conserved quantity that is preserved in the associated numerical methods.

The structure-preservingproperty of numerical method is a very important factor to estimate the computational efficiency of the numerical scheme in the geometric integration theory ^{[1, 2, 3, 4, 5]} . During the last three decades,there are many significant works reported on the structure- preserving properties on several kinds of geometric integration methods,e.g.,the symplectic algorithm ^{[1, 2]} ,the Lie group algorithm ^[6] ,the multi-symplectic integrator ^{[5, 7, 8, 9]} ,and the gener- alized multi-symplectic method ^{[4, 10]} .

All of the geometric integration methods mentioned above devote to preserving the inherent geometric properties of the systems. To realize this purpose,the constructed differential scheme must own some structure-preserving properties on the time space or manifold. In this sense, the structure-preserving property of the scheme is the soul of the geometric integration method.

In this paper,the structure-preserving properties of three classic differential schemes for the Hamiltonian system are discussed in detail,and the numerical results of some numerical experiments on the oscillator system are presented to illustrate the structure-preserving properties of the three classic differential schemes.

2 Three differential schemes of Hamiltonian system

In classic mechanics,we always consider a mechanical system with the generalized coordi- nates q ∈ R^d and the Lagrangian function L = T − V ,where T (T ≡ T(q,˙q)) is the system kinetic energy and V (V ≡ V (q)) is the system potential energy. The motion of this mechanical system can be described as the following Euler-Lagrange equation:

Introduce an intermediate variable

Define the Hamiltonian function via the Legendre transformation as follows:

Then,the Euler-Lagrange system (1) can be translated into the Hamiltonian canonical form as follows: It is worth mentioning that H (H ≡ T + V ) represents the total energy of the mechanical system and it is a conserved quantity if the system is conservative.

More generally,define the state variable as

Then,Eq.(2) can be rewritten as a more compact form ^[11] as follows: where J is the skew-symplectic matrix expressed by

and I_d is the identity matrix of the dimension d.

In this paper,the following three classic differential schemes for the Hamiltonian system (2) are considered:

(i) Standard forward Euler scheme ^[12]

(ii) Symplectic Euler scheme ^[2]

(iii) Störmer-Verlet scheme ^[13]

In the above equations,h is the step length.

Obviouslly,all of these three schemes are explicit. The standard forward Euler scheme is an ancient numerical scheme for generalized numerical problems. Both the symplectic Euler scheme and the Störmer-Verlet scheme are numerical schemes resulting from the symplectic geometry theory.

3 Structure-preserving properties of three differential schemes

In this section,the structure-preserving properties of the three differential schemes mentioned above for a classic Hamiltonian system will be discussed theoretically.

Consider the following oscillator:

where q and p are scalars.

The Hamiltonian function (total energy) of Eq.(8) is

which is an ellipse curve.

3.1 Structure-preserving properties of forward Euler scheme

The forward Euler scheme of the oscillator (8) is

Let ψ_h be the discrete evolutionary operator given by

From this scheme,we can easily get the recurrence relationship associated with the Hamiltonian function as follows:

which implies that the Hamiltonian function is not preserved obviously.

3.2 Structure-preserving properties of symplectic Euler scheme

The symplectic Euler scheme of the oscillator (8) is

The discrete evolutionary operator is

Following the outline above,the recurrence relationship associated with the Hamiltonian function can be obtained as follows:

Then,we can get that the ellipse map (9) can be now mapped to Obviously,this modified Hamiltonian function is an ellipse curve that is converged to the ellipse curve given by Eq.(9) if h → 0,which implies the symplectic Euler scheme (6). The error between the modified Hamiltonian function H and the Hamiltonian function H will increase when the step length h increases.

3.3 Structure-preserving properties of Störmer-Verlet scheme

The Störmer-Verlet schemes of the oscillator (8) are

The discrete evolutionary operator can be written as Following the outline above,the recurrence relationship can be obtained as follows: which implies

Therefore,the Störmer-Verlet scheme can preserve the total energy of the oscillator system (8) exactly.

4 Numerical experiments

To illustrate the structure-preserving properties of the three differential schemes for the oscillator system,several numerical experiments are performed under the same initial value (p,q)^T|_t=0 = (0,1)^T in this section.

Let h = 0.05s. The phase diagram of the oscillator system (8) obtained by the standard forward Euler scheme (10) is shown in Fig. 1. From Fig. 1,it can be found that the orbit of the oscillator obtained by the standard forward Euler scheme (10) is not closed,i.e.,the orbit is nonperiodic,which implies that the standard forward Euler scheme (10) cannot preserve the periodicity and boundedness of the oscillator system. This result agrees with Eq.(12). This is the intrinsic reason why the orbit obtained by the standard forward Euler scheme (10) is not so closed and why the standard forward Euler scheme (10) cannot preserve any (modified) Hamiltonian function of the oscillator system at all.

The relative error of the Hamiltonian function (9) associated with the oscillator system is shown in Fig. 2. From Fig. 2,it can be found that using the standard forward Euler scheme, the relative error of the Hamiltonian function (9) increases linearly with the time elapsing approximately; the maximum relative error of the Hamiltonian functions is about 25% when t = 960s. These results agree well with the recurrence relationship (12). This is also the reason why the radius of the curvature of the orbit obtained by the standard forward Euler scheme does not increase linearly.

To illustrate the structure-preserving properties of the symplectic Euler scheme (14) and to investigate the relationship between the structure-preserving properties of the symplectic Euler scheme and the step length,let h = 0.05s,0.1s,and 0.5s. The phase diagrams of the oscillator system (8) obtained by the symplectic Euler scheme (14) with different step lengths are shown in Fig. 3. In Fig. 3,the orbits obtained by the symplectic Euler scheme (14) are closed and periodic,which implies that the symplectic Euler scheme (14) can preserve a certain Hamiltonian function. However,the orbits obtained by the symplectic Euler scheme (14) are not coincided with the original periodic orbit and diverge from the original periodic orbit farther with the increase in the step length.

The above phenomena are resulted from the errors between the modified Hamiltonian function H and the Hamiltonian function H,which can be further concluded from the relative errors for the Hamiltonian function with different step lengths (see Fig. 4(a)-(c)). From Fig. 4(a)-(c),it can be found that the relative errors for the Hamiltonian function of the symplectic Euler scheme increase with the increase in the step length. The maximum relative error for the Hamiltonian function is about 1.9E−5 when h = 0.05 s,the maximum relative error is about 9.8E−5 when h = 0.1 s,and the maximum relative error is about 3.2E−3 when h = 0.5 s.

For the Störmer-Verlet scheme,the phase diagram of the oscillator system with h = 0.5 s is shown in Fig. 5. From Fig. 5,it can be found that the orbit obtained by the StörmerVerlet scheme is well coincided with the original periodic orbit. The relative error for the Hamiltonian function with h = 0.5 s is shown in Fig. 4(d),where the maximum relative error for the Hamiltonian function with h = 0.5 s is about 1.5E−5.

Comparing the results obtained by the standard forward Euler scheme,the symplectic Euler scheme,and the Störmer-Verlet scheme,we can conclude that the standard forward Euler scheme cannot preserve any structure of the oscillator system,and the relative error for the Hamiltonian function is large even if the step length is small; the symplectic Euler scheme can preserve the periodicity and the boundedness of the oscillator system,but cannot preserve the Hamiltonian function exactly; the Störmer-Verlet scheme can preserve the periodicity,the boundedness,and the Hamiltonian function of the oscillator system exactly.

5 Conclusions

The structure-preserving property is an important factor used to assess the effect of a nu- merical method. In this paper,the structure-preserving properties of the three differential schemes for an oscillator system,i.e.,the standard forward Euler scheme,the symplectic Euler scheme,and the Störmer-Verlet scheme,are discussed,respectively. The numerical results of these differential schemes verify the theoretical results in this paper,and illustrate the following structure-preserving properties:

(i) The standard forward Euler scheme cannot preserve any structure-preserving property of the Hamiltonian system.

(ii) The symplectic Euler scheme can preserve two structure-preserving properties of the Hamiltonian system.

(iii) The Störmer-Verlet scheme can preserve the three structure-preserving properties of the Hamiltonian system.

These conclusions give some advices on the selection of the difference discrete method for the Hamiltonian system.

and the numerical