An Energy-Work Relationship Integration Scheme for Nonconservative Hamiltonian Systems

This letter focuses on studying a new energy-work relationship numerical integration scheme of nonconservative Hamiltonian systems. The signal-stage, multistage, and parallel composition numerical integration schemes are presented for this system. The high-order energy-work relation scheme of the system is constructed by a parallel connection of n multistage scheme of order 2 which its order of accuracy is 2n. The connection, which is discrete analog of usual case, between the change of energy and work of nonconservative force is obtained for nonconservative Hamiltonian systems. This letter also shows that the more the stages of the schemes are, the less the error rate of the scheme is for nonconservative Hamiltonian systems. Finally, an applied example is discussed to illustrate these results.


INTRODUCTION
Recently, there have been a great number of studies on the so-called geometric numerical integration scheme which preserve the structure of systems [1][2][3].Leimkuher and Reich pointed out that the geometric numerical integrators are time-stepping methods, designed such that they exactly satisfy conservation laws, symmetries or symplectic properties of a system of differential equations [1].Hairer et al. presented the symplectic integration of separable Hamiltonian ordinary and partial differential equations.In this way, the symplectic scheme is performed prior to the spatial step as opposed to the standard approach of spatially discrediting the PDE to form a system of Hamiltonian ODEs to which a symplectic integrator can be applied [2].An energyconserving scheme is one of such geometric numerical integration scheme [4][5][6][7][8].It is very known that a highorder scheme can be constructed by connecting low-order scheme in series (hereafter we will call it series composition) [1][2][3].Now, the high-order energy-conserving scheme has been constructed with method [9].In this letter, we will present a new numerical integration scheme, which is energy-work relation integration scheme, of nonconservative Hamiltonian systems.This work also study that a highorder energy-work relation scheme, which it has a structure connecting the order 2 multistage scheme in parallel scheme (hereafter we will call it parallel composition scheme), can be constructed by connecting low-order scheme in series.

NUMERICAL INTEGRATION FOR NONCONSERVATIVE HAMILTONIAN SYSTEMS
Let the configuration of a mechanical system is described by n generalized coordinates q s (s = 1, . . ., n) and n generalized momentums p s (s = 1, . . ., n).Suppose the system is subjected to n nonpotential generalized forces Q s .The generalized Hamiltonian canonical equations of system as here, the Hamiltonian which represents the total energy.The relationship between the change of energy and the power of nonconservative force is easily verified as The numerical integration is considered as the discretization of ∂H q 1 , . . ., q n , p 1 , . . ., p n ∂p s dt, ∂H q 1 , . . ., q n , p 1 , . . ., p n ∂q s dt+Q s Δt, which are obtained by integrating both sides of (1) on the interval [t k , t k+1 ], where Δt is the step size.

Single-stage scheme
Let p k s and q k s be the numerical approximations of p s (t k ) and q s (t k ), respectively.Then a 1-stage scheme is given by with The notations δ a,b ps and δ a,b qs denote the partial difference quotient operators with respect to p s and q s , respectively, which are defined as δ a,b qs F q l1 1 , . . ., q ln n , p k1 1 , . . ., where E a qs and E a ps are the shift operators defined as E a qs F q l1 1 , . . ., q ls−1 s−1 , q ls s , q ls+1 s+1 , . . ., q ln n , p k1 1 , . . ., p kn n = F q l1 1 , . . ., q ls−1 s−1 , q ls+a s , q ls+1 s+1 , . . ., q ln n , p k1 1 , . . ., p kn n , The notations μ a,b qs and μ a,b ps denote the mean difference operators with respect to all variables except for q s and p s , respectively, which are defined as with where per(A) denotes the permanent or plus determinant of a matrix A [10].For example, in the case d = 1, we have (11) The operators δ a,b qs , δ a,b ps , μ a,b qs , and μ a,b ps have symmetry expressed as (12)

Relation between the energy and work of nonconservative force for nonconservative Hamiltonian systems
Proposition 1.The relation between the energy and work of nonconservative force for nonconservative Hamiltonian system holds: Proof.For simplicity, we set We first note the identity It follows that where we have used the properties of the permanent and the definitions of operators ( 7)-( 10) [10].
Proposition 2. The scheme {(1), . . ., (6)} satisfies the relation between the change of energy and the work of nonconservative force for the system.
Proof.We see from the chain rule (13) that the change of energy H k is equivalent to the work of nonconservative force: which is a discrete analog of (3).

Order of accuracy
The local errors involved in the determination of {p k+1 s , , that is, I 1,0 qs and I 1,0 ps in the scheme ( 5) are the second-order approximations of the integrals in (4), respectively.Although this can be proved by the Taylor expansions, it is obvious because the scheme is symmetric (see Section 4.3).

Multistage scheme
An c-stage scheme is constructed by connecting the secondorder scheme with small integration interval of length Δt/c in series: with where P k+m/c s and Q k+m/c s are the internal stagevariables.It should be noted that the above scheme is equivalent to the scheme: Ps , The latter scheme (20) will be used in the next section to construct a higher-order scheme.
It is obvious for the c-stage that the relationship between the change of energy and the work of nonconservative force is exactly equivalent and that the order of accuracy is 2. We point out here that the local error is expressed as

Parallel composition scheme
Let c 1 , c 2 , . . ., c n be arbitrary positive integers satisfying then a new scheme is constructed by connection c 1 -stage, c 2 -stage, . . ., c n -stage schemes of order 2 in parallel: with the weights where (24)

Relation between the change of energy and work of nonconservative force
Proposition 3. The scheme (23) with the condition

satisfies relationship between the change of energy and work of nonconservative force for nonconservative Hamiltonian systems
Proof.We first note We see from Proposition 1 that It follows from ( 25)-( 27) that  29) into (28) yields which is a discrete analog of that relation between the change of energy and work of nonconservative force for the systems (17).

Order of a symmetric scheme
Proposition 4. Consider the scheme (22) as mapping and let φ −1  Δt be the inverse mapping of φ Δt .Then, one has That is, the scheme is symmetric Proof.The inverse φ −1 Δt is obtained by exchanging (p k s , q k s ) and (p k+1 s , q k+1 s ).Replacing Δt by −Δt and rearranging terms in φ −1  Δt leads to the mapping φ −1 −Δt .For this φ −1 −Δt , setting and omitting the tilde, we can obtain φ Δt .Therefore, form Proof.It is known that if a one-step scheme is symmetric, its order of accuracy is even [1,2].Therefore, the local error of the scheme φ Δt is O(Δt 2r+1 ) with a positive integer r.We first choose {d j } u j=1 such that , and then the O(Δt 3 )-term in the error of φ Δt vanishes.Since the error of φ Δt is of odd order, it becomes O(Δt 5 ).The O(Δt 5 )term in the error {d j } u j=1 -term in the error of φ Δt is expressed as These procedures can be repeated.The final condition for Therefore, if one chooses {d j } u j=1 such that they satisfy the n simultaneous linear equations: then theerror of φ Δt is O(Δt 2u+1 ).Since the solution of (39) is given by ( 23), the order of accuracy is 2u.

A NUMERICAL EXAMPLE
Consider the motion of a particle with unit mass whose Hamiltonian is and the motion of the system is subjected to nonpotential force where b is a constant.The equation of motion of nonconservative particle is The analytic solution of (42) is given by which have the period T = 2π/b.We take the initial conditions: and the calculation time t = T.The parallel composition scheme with c j = j, j = 1, 2, . . ., u (45) was used.We calculated the global error given by e(t) = p K − p(T) where K = T/Δt.Since the global error e(t) is about T/Δt times the local error, e(t) is expressed as e(T) = O(Δt 2n ).We should point out that the local error of the parallel composition is expressed as the more the stages of the schemes are, the smaller the error of the scheme for nonconservative Hamiltonian systems.

CONCLUSION
In this paper, the new numerical integration schemes of nonconservative Hamilton systems are established.This study has given that the numerical connection between energy of system and work of nonconservative force is an analog of usual energy-work connection, and the numerical connection between the high-order energy-work is also contented.Numerical results showed that the more the stages of the schemes are, the smaller the error of the scheme for nonconservative Hamiltonian systems.

Proposition 5 .
If one chooses the weights d 1 , d 2 , . . ., d n as (23), the accuracy of the scheme (22) is at least of order 2n.