Abstract

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.

1. Introduction

Recently, there have been a great number of studies on the so-called geometric numerical integration scheme which preserve the structure of systems [13]. 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 energy-conserving scheme is one of such geometric numerical integration scheme [48]. It is very known that a high-order scheme can be constructed by connecting low-order scheme in series (hereafter we will call it series composition) [13]. 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 high-order 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.

2. Numerical Integration for Nonconservative Hamiltonian Systems

Let the configuration of a mechanical system is described by n generalized coordinates 𝑞𝑠(𝑠=1,,𝑛) and n gener-alized momentums 𝑝𝑠(𝑠=1,,𝑛). Suppose the system is subjected to n nonpotential generalized forces 𝑄𝑠. The gener-alized Hamiltonian canonical equations of system as ̇𝑞𝑠=𝜕𝐻𝜕𝑝𝑠,̇𝑝𝑠=𝜕𝐻𝜕𝑞𝑠+𝑄𝑠,𝑠=1,2,,𝑛,(1) here, the Hamiltonian 𝐻=𝐻𝑞1,,𝑞𝑛,𝑝1,,𝑝𝑛,(2) which represents the total energy. The relationship between the change of energy and the power of nonconservative force is easily verified as 𝑑𝐻𝑑𝑡=𝑛𝑠=1𝜕𝐻𝜕𝑞𝑠̇𝑞𝑠+𝜕𝐻𝜕𝑝𝑠̇𝑝𝑠=𝑛𝑠=1𝜕𝐻𝜕𝑞𝑠𝜕𝐻𝜕𝑝𝑠𝜕𝐻𝜕𝑝𝑠𝜕𝐻𝜕𝑞𝑠+𝜕𝐻𝜕𝑝𝑠𝑄𝑠=𝑛𝑠=1𝜕𝐻𝜕𝑝𝑠𝑄𝑠.(3) The numerical integration is considered as the discretization of 𝑞𝑠𝑡𝑘+1=𝑞𝑠𝑡𝑘+𝑡𝑘+1𝑡𝑘𝜕𝐻𝑞1,,𝑞𝑛,𝑝1,,𝑝𝑛𝜕𝑝𝑠𝑑𝑡,𝑝𝑠𝑡𝑘+1=𝑝𝑠𝑡𝑘𝑡𝑘+1𝑡𝑘𝜕𝐻𝑞1,,𝑞𝑛,𝑝1,,𝑝𝑛𝜕𝑞𝑠𝑑𝑡+𝑄𝑠Δ𝑡,𝑡𝑘=𝑘Δ𝑡,𝑘=0,1,2,,(4) which are obtained by integrating both sides of (1) on the interval [𝑡𝑘,𝑡𝑘+1], where Δ𝑡 is the step size.

3. Second-Order Schemes of Numerical Integration for Nonconservative Hamiltonian Systems

3.1. Single-Stage Scheme

Let 𝑝𝑘𝑠 and 𝑞𝑘𝑠 be the numerical approximations of 𝑝𝑠(𝑡𝑘) and 𝑞𝑠(𝑡𝑘), respectively. Then a 1-stage scheme is given by

𝑞𝑘+1𝑠=𝑞𝑘𝑠+𝐼1,0𝑝𝑠,𝑝𝑘+1𝑠=𝑝𝑘𝑠𝐼1,0𝑞𝑠+𝑄𝑠Δ𝑡,𝑠=1,2,,𝑛,(5) with

𝐼𝑎,𝑏𝑞𝑠=𝑎𝑏Δ𝑡𝛿𝑎,𝑏𝑞𝑠𝜇𝑎,𝑏𝑞𝑠𝐻𝑘,𝐼𝑎,𝑏𝑝𝑠=𝑎𝑏Δ𝑡𝛿𝑎,𝑏𝑝𝑠𝜇𝑎,𝑏𝑝𝑠𝐻𝑘,𝐻𝑘=𝐻𝑞𝑘1,,𝑞𝑘𝑛,𝑝𝑘1,,𝑝𝑘𝑛.(6) The notations 𝛿𝑎,𝑏𝑝𝑠 and 𝛿𝑎,𝑏𝑞𝑠 denote the partial difference quotient operators with respect to 𝑝𝑠 and 𝑞𝑠, respectively, which are defined as

𝛿𝑎,𝑏𝑞𝑠𝐹𝑞𝑙11,,𝑞𝑙𝑛𝑛,𝑝𝑘11,,𝑝𝑘𝑛𝑛=𝐸𝑎𝑞𝑠𝐸𝑏𝑞𝑠𝐹𝐸𝑎𝑞𝑠𝐸𝑏𝑞𝑠𝑞𝑙𝑠𝑠,𝛿𝑎,𝑏𝑝𝑠𝐹𝑞𝑙11,,𝑞𝑙𝑛𝑛,𝑝𝑘11,,𝑝𝑘𝑛𝑛=𝐸𝑎𝑝𝑠𝐸𝑏𝑝𝑠𝐹𝐸𝑎𝑝𝑠𝐸𝑏𝑝𝑠𝑝𝑘𝑠𝑠,(7) where 𝐸𝑎𝑞𝑠 and 𝐸𝑎𝑝𝑠 are the shift operators defined as

𝐸𝑎𝑞𝑠𝐹𝑞𝑙11,,𝑞𝑙𝑠1𝑠1,𝑞𝑙𝑠𝑠,𝑞𝑙𝑠+1𝑠+1,,𝑞𝑙𝑛𝑛,𝑝𝑘11,,𝑝𝑘𝑛𝑛=𝐹𝑞𝑙11,,𝑞𝑙𝑠1𝑠1,𝑞𝑙𝑠+𝑎𝑠,𝑞𝑙𝑠+1𝑠+1,,𝑞𝑙𝑛𝑛,𝑝𝑘11,,𝑝𝑘𝑛𝑛,𝐸𝑎𝑝𝑠𝐹𝑞𝑙11,,𝑞𝑙𝑛𝑛,𝑝𝑘11,,𝑝𝑘𝑠1𝑠1,𝑝𝑘𝑠𝑠,𝑝𝑘𝑠+1𝑠+1,,𝑝𝑘𝑛𝑛=𝐹𝑞𝑙11,,𝑞𝑙𝑛𝑛,𝑝𝑘11,,𝑝𝑘𝑠1𝑠1,𝑝𝑘𝑠+𝑎𝑠,𝑝𝑘𝑠+1𝑠+1,,𝑝𝑘𝑛𝑛.(8) The notations 𝜇𝑎,𝑏𝑞𝑠 and 𝜇𝑎,𝑏𝑝𝑠 denote the mean difference operators with respect to all variables except for 𝑞𝑠 and 𝑝𝑠, respectively, which are defined as

𝜇𝑎,𝑏𝑞𝑠𝐹𝑞𝑙11,,𝑞𝑙𝑛𝑛,𝑝𝑘11,,𝑝𝑘𝑛𝑛=𝑀𝑎,𝑏𝐸𝑞1,,𝐸𝑞𝑠1,𝐸𝑞𝑠+1,,𝐸𝑞𝑛,𝐸𝑝1,,𝐸𝑝𝑛,𝜇𝑎,𝑏𝑞𝑠𝐹𝑞𝑙11,,𝑞𝑙𝑛𝑛,𝑝𝑘11,,𝑝𝑘𝑛𝑛=𝑀𝑎,𝑏𝐸𝑞1,,𝐸𝑞𝑛,𝐸𝑝1,,𝐸𝑝𝑠1,𝐸𝑝𝑠+1,,𝐸𝑝𝑛,(9) with

𝑀𝑎,𝑏𝑥1,𝑥2,,𝑥𝑟1=1𝑟!𝑟𝑙=1per𝑥𝑎1𝑥𝑎2𝑥𝑎𝑟1𝑥𝑎1𝑥𝑎2𝑥𝑎𝑟1𝑥𝑏1𝑥𝑏2𝑥𝑏𝑟1𝑥𝑏1𝑥𝑏2𝑥𝑏𝑟125𝑝𝑡}𝑟125𝑝𝑡}𝑙1,(10) where per(A) denotes the permanent or plus determinant of a matrix A [10]. For example, in the case d = 1, we have

𝜇𝑎,𝑏𝑞1=𝑀𝑎,𝑏𝐸𝑞1=12𝐸𝑎𝑞1+𝐸𝑏𝑞1,𝜇𝑎,𝑏𝑝1=𝑀𝑎,𝑏𝐸𝑝1=12𝐸𝑎𝑝1+𝐸𝑏𝑝1.(11) The operators 𝛿𝑎,𝑏𝑞𝑠,𝛿𝑎,𝑏𝑝𝑠,𝜇𝑎,𝑏𝑞𝑠, and 𝜇𝑎,𝑏𝑝𝑠 have symmetry expressed as

𝛿𝑎,𝑏𝑞𝑠=𝛿𝑏,𝑎𝑞𝑠,𝛿𝑎,𝑏𝑝𝑠=𝛿𝑏,𝑎𝑝𝑠,𝜇𝑎,𝑏𝑞𝑠=𝜇𝑏,𝑎𝑞𝑠,𝜇𝑎,𝑏𝑝𝑠=𝜇𝑏,𝑎𝑝𝑠.(12)

3.2. 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:
𝐻𝑘+𝑎𝐻𝑘+𝑏=𝑛𝑠=1𝛿𝑎,𝑏𝑞𝑠𝜇𝑎,𝑏𝑞𝑠𝐻𝑘𝑞𝑘+𝑎𝑠𝑞𝑘+𝑏𝑠+𝛿𝑎,𝑏𝑝𝑠𝜇𝑎,𝑏𝑝𝑠𝐻𝑘𝑝𝑘+𝑎𝑠𝑝𝑘+𝑏𝑠.(13)

Proof. For simplicity, we set
𝐸𝑎=𝐸𝑎𝑞1,,𝐸𝑎𝑞𝑛,𝐸𝑎𝑝1,,𝐸𝑎𝑝𝑛𝑇.(14) We first note the identity
𝑛𝑠𝐸𝑎𝑞𝑠𝐸𝑎𝑝𝑠=12𝑛!per(𝐸𝑎,,𝐸𝑎2𝑛).(15) It follows that 𝐻𝑘+𝑎𝐻𝑘+𝑏=(𝑛𝑠=1𝐸𝑎𝑞𝑠𝐸𝑎𝑝𝑠𝑛𝑠=1𝐸𝑏𝑞𝑠𝐸𝑏𝑝𝑠)𝐻𝑘=𝑛𝑠=1𝐸𝑎𝑞𝑠𝐸𝑏𝑞𝑠𝑀𝑎,𝑏𝐸𝑞1,,𝐸𝑞𝑛,𝐸𝑝1,,𝐸𝑝𝑠1,𝐸𝑝𝑠+1,,𝐸𝑝𝑛+𝐸𝑎𝑝𝑠𝐸𝑏𝑝𝑠𝑀𝑎,𝑏𝐸𝑞1,,𝐸𝑞𝑠1,𝐸𝑞𝑠+1,,𝐸𝑞𝑛,𝐸𝑝1,,𝐸𝑝𝑛𝐻𝑘=𝑛𝑠=1𝛿𝑎,𝑏𝑞𝑠𝜇𝑎,𝑏𝑞𝑠𝐻𝑘𝑞𝑘+𝑎𝑠𝑞𝑘+𝑏𝑠+𝛿𝑎,𝑏𝑝𝑠𝜇𝑎,𝑏𝑝𝑠𝐻𝑘𝑝𝑘+𝑎𝑠𝑝𝑘+𝑏𝑠,(16) 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 𝐻𝑘 is equivalent to the work of nonconservative force:
𝐻𝑘+1𝐻𝑘=𝑛𝑠=1𝛿1,0𝑞𝑠𝜇1,0𝑞𝑠𝐻𝑘𝑞𝑘+1𝑠𝑞𝑘𝑠+𝛿1,0𝑝𝑠𝜇1,0𝑝𝑠𝐻𝑘𝑝𝑘+1𝑠𝑝𝑘𝑠=Δ𝑡𝑛𝑠=1𝛿1,0𝑞𝑠𝜇1,0𝑞𝑠𝐻𝑘𝛿1,0𝑝𝑠𝜇1,0𝑝𝑠𝐻𝑘+𝛿1,0𝑝𝑠𝜇1,0𝑝𝑠𝐻𝑘𝛿1,0𝑞𝑠𝜇1,0𝑞𝑠𝐻𝑘+𝑄𝑠=𝑛𝑠=1Δ𝑡𝛿1,0𝑝𝑠𝜇1,0𝑝𝑠𝐻𝑘𝑄𝑠,(17) which is a discrete analog of (3).

3.3. Order of Accuracy

The local errors involved in the determination of {𝑝𝑘+1𝑠,𝑞𝑘+1𝑠}𝑛𝑠=1 from {𝑝𝑘𝑠,𝑞𝑘𝑠}𝑛𝑠=1 are 𝑂(Δ𝑡3), that is, 𝐼1,0𝑞𝑠 and 𝐼1,0𝑝𝑠 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).

3.4. Multistage Scheme

An c-stage scheme is constructed by connecting the second-order scheme with small integration interval of length Δ𝑡/𝑐 in series:

𝑃𝑘+𝑚/𝑐𝑠=𝑃𝑘+(𝑚1)/𝑐𝑠𝐼𝑚/𝑐,(𝑚1)/𝑐𝑄𝑠+𝑄𝑠𝑄𝑘1,,𝑄𝑘𝑛,𝑃𝑘1,,𝑃𝑘𝑛Δ𝑡𝑐,𝑄𝑘+𝑚/𝑐𝑠=𝑄𝑘+(𝑚1)/𝑐𝑠+𝐼𝑚/𝑐,(𝑚1)/𝑐𝑃𝑠𝑃𝑘+1𝑠𝑝𝑘+1𝑠,𝑃𝑘𝑠=𝑝𝑘𝑠,𝑄𝑘+1𝑠=𝑞𝑘+1𝑠,𝑄𝑘𝑠=𝑞𝑘𝑠,𝑠=1,2,,𝑛,𝑚=1,2,,𝑐,(18) with

𝐼𝑎,𝑏𝑄𝑠=𝑎𝑏Δ𝑡𝛿𝑎,𝑏𝑄𝑠𝜇𝑎,𝑏𝑄𝑠𝐻𝑄𝑘1,,𝑄𝑘𝑛,𝑃𝑘1,,𝑃𝑘𝑛,𝐼𝑎,𝑏𝑃𝑠=𝑎𝑏Δ𝑡𝛿𝑎,𝑏𝑃𝑠𝜇𝑎,𝑏𝑃𝑠𝐻𝑄𝑘1,,𝑄𝑘𝑛,𝑃𝑘1,,𝑃𝑘𝑛,(19) where 𝑃𝑘+𝑚/𝑐𝑠 and 𝑄𝑘+𝑚/𝑐𝑠 are the internal stage variables. It should be noted that the above scheme is equivalent to the scheme:

𝑝𝑘+1𝑠=𝑝𝑘𝑠𝑐𝑙=1𝐼𝑙/𝑐,(𝑙1)/𝑐𝑄𝑠+𝑄𝑠𝑞𝑘1,,𝑞𝑘𝑛,𝑝𝑘1,,𝑝𝑘𝑛Δ𝑡𝑐,𝑞𝑘+1𝑠=𝑞𝑘𝑠𝑐𝑙=1𝐼𝑙/𝑐,(𝑙1)/𝑐𝑃𝑠,𝑃𝑘+𝑚/𝑐𝑠=𝑐𝑚𝑐(𝑝𝑘𝑠𝑚𝑙=1𝐼𝑙/𝑐,(𝑙1)/𝑐𝑄𝑠+𝑄𝑠Δ𝑡𝑐)+𝑚𝑐(𝑝𝑘+1𝑠+𝑐𝑙=𝑚+1𝐼𝑙/𝑐,(𝑙1)/𝑐𝑄𝑠+𝑄𝑠Δ𝑡𝑐),𝑄𝑘+𝑚/𝑐𝑠=𝑐𝑚𝑐(𝑞𝑘𝑠+𝑚𝑙=1𝐼𝑙/𝑐,(𝑙1)/𝑐𝑃𝑠)+𝑚𝑐(𝑞𝑘+1𝑠𝑐𝑙=𝑚+1𝐼𝑙/𝑐,(𝑙1)/𝑐𝑃𝑠),𝑠=1,2,,𝑛,𝑚=1,2,,𝑐1.(20) 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 𝑐×𝑂[(Δ𝑡/𝑐)3]=𝑐2𝑂(Δ𝑡)3.

4. Higher-Order Schemes of Numerical Integration for Nonconservative Hamiltonian Systems

4.1. Parallel Composition Scheme

Let 𝑐1,𝑐2,,𝑐𝑛 be arbitrary positive integers satisfying

𝑐1<𝑐2<<𝑐𝑛,(21) then a new scheme is constructed by connection 𝑐1-stage,𝑐2-stage,,𝑐𝑛-stage schemes of order 2 in parallel: 𝑝𝑘+1𝑠=𝑝𝑘𝑠𝑢𝑗=1𝑑𝑗𝑐𝑗𝑙=1𝐼𝑙/𝑐𝑗,(𝑙1)/𝑐𝑗𝑄𝑠𝑗+𝑄𝑠𝑞𝑘1,,𝑞𝑘𝑛,𝑝𝑘1,,𝑝𝑘𝑛Δ𝑡,𝑞𝑘+1𝑠=𝑞𝑘𝑠+𝑢𝑗=1𝑑𝑗𝑐𝑗𝑙=1𝐼𝑙/𝑐𝑗,(𝑙1)/𝑐𝑗𝑃𝑖𝑗,𝑃𝑘+𝑚/𝑐𝑗𝑠𝑗=𝑐𝑗𝑚𝑐𝑗(𝑝𝑘𝑠𝑚𝑙=1𝐼𝑙/𝑐𝑗,(𝑙1)/𝑐𝑗𝑄𝑠𝑗+𝑄𝑠𝑗Δ𝑡)+𝑚𝑐𝑗(𝑝𝑘+1𝑠𝑚𝑙=𝑚+1𝐼𝑙/𝑐𝑗,(𝑙1)/𝑐𝑗𝑄𝑠𝑗+𝑄𝑠𝑗Δ𝑡),𝑄𝑘+𝑚/𝑐𝑗𝑠𝑗=𝑐𝑗𝑚𝑐𝑗(𝑞𝑘𝑠+𝑚𝑙=1𝐼𝑙/𝑐𝑗,(𝑙1)/𝑐𝑗𝑄𝑠𝑗)+𝑚𝑐𝑗(𝑞𝑘+1𝑠+𝑚𝑙=𝑚+1𝐼𝑙/𝑐𝑗,(𝑙1)/𝑐𝑗𝑄𝑠𝑗),𝑃𝑘+1𝑠𝑗=𝑝𝑘+1𝑠,𝑃𝑘𝑠𝑗=𝑝𝑘𝑠,𝑄𝑘+1𝑠𝑗=𝑞𝑘+1𝑠,𝑄𝑘𝑠𝑗=𝑞𝑘𝑠,𝑠=1,2,,𝑛,𝑗=1,2,,𝑢,𝑚=1,2,,𝑐𝑗1,(22) with the weights

𝑑𝑗=1for𝑢=1,𝑐2𝑢2𝑗𝑢𝑙=1,𝑙𝑗𝑐2𝑗𝑐2𝑙for𝑢2,𝑗=1,2,,𝑢,(23) where

𝐼𝑎,𝑏𝑃𝑠𝑗=𝑎𝑏Δ𝑡𝛿𝑎,𝑏𝑃𝑠𝑗𝜇𝑎,𝑏𝑃𝑠𝑗𝐻𝑘𝑗,𝐼𝑎,𝑏𝑄𝑠𝑗=𝑎𝑏Δ𝑡𝛿𝑎,𝑏𝑄𝑠𝑗𝜇𝑎,𝑏𝑄𝑠𝑗𝐻𝑘𝑗,𝐻𝑘𝑗=𝐻𝑄𝑘1𝑗,,𝑄𝑘𝑛𝑗,𝑃𝑘1𝑗,,𝑃𝑘𝑛𝑗,Λ𝑘𝑠𝑗=Λ𝑠𝑄𝑘1𝑗,,𝑄𝑘𝑛𝑗,𝑃𝑘1𝑗,,𝑃𝑘𝑛𝑗.(24)

4.2. Relation between the Change of Energy and Work of Nonconservative Force

Proposition 3. The scheme (23) with the condition
𝑢𝑗=1𝑑𝑗=1(25) satisfies relationship between the change of energy and work of nonconservative force for nonconservative Hamiltonian systems

Proof. We first note
𝐻𝑘+1=𝐻𝑘+1𝑗,𝐻𝑘=𝐻𝑘𝑗,𝑗=1,2,,𝑢.(26) We see from Proposition 1 that
𝐻𝑘+𝑎𝑗𝐻𝑘+𝑏𝑗=𝑛𝑠=1𝛿𝑎,𝑏𝑃𝑠𝑗𝜇𝑎,𝑏𝑃𝑠𝑗𝐻𝑘𝑗𝑃𝑘+𝑎𝑠𝑗𝑃𝑘+𝑏𝑠𝑗+𝛿𝑎,𝑏𝑄𝑠𝑗𝜇𝑎,𝑏𝑄𝑠𝑗𝐻𝑘𝑗𝑄𝑘+𝑎𝑠𝑗𝑄𝑘+𝑏𝑠𝑗.(27) It follows from (25)–(27) that
𝐻𝑘+1𝐻𝑘=𝑢𝑗=1𝑑𝑗𝐻𝑘+1𝑗𝐻𝑘𝑗=𝑢𝑗=1𝑑𝑗𝑐𝑗𝑚=1𝐻𝑘+𝑚/𝑐𝑗𝑗𝐻𝑘+(𝑚1)/𝑐𝑗𝑗=𝑢𝑗=1𝑑𝑗𝑐𝑗Δ𝑡𝑐𝑗𝑚=1𝑛𝑠=1𝐼𝑘+𝑚/𝑐𝑗,𝑘+(𝑚1)/𝑐𝑗𝑃𝑠𝑗×𝑃𝑘+𝑚/𝑐𝑗𝑠𝑗𝑃𝑘+(𝑚1)/𝑐𝑗𝑠𝑗+𝐼𝑘+𝑚/𝑐𝑗,𝑘+(𝑚1)/𝑐𝑗𝑄𝑠𝑗×𝑄𝑘+𝑚/𝑐𝑗𝑠𝑗𝑄𝑘+(𝑚1)/𝑐𝑗𝑠𝑗.(28) We obtain from (22)
𝑃𝑘+𝑚/𝑐𝑗𝑠𝑗𝑃𝑘+(𝑚1)/𝑐𝑗𝑠𝑗=1𝑐𝑗(𝑢𝑟=1𝑐𝑟𝑐𝑗𝑙=1𝐼𝑙/𝑐𝑟,(𝑙1)/𝑐𝑟𝑃𝑠𝑟𝑐𝑗𝑙=1𝐼𝑙/𝑐𝑗,(𝑙1)/𝑐𝑗𝑃𝑠𝑗)+𝐼𝑚/𝑐𝑗,(𝑚1)/𝑐𝑗𝑃𝑠𝑗(29) Substituting (29) into (28) yields
𝐻𝑘+1𝐻𝑘=𝑢𝑗=1𝑑𝑗𝑐𝑗𝑚=1𝑐𝑗𝑛𝑠=1𝐼𝑚/𝑐𝑗,(𝑚1)/𝑐𝑗𝑃𝑠𝑗𝑄𝑠𝑗,(30) which is a discrete analog of that relation between the change of energy and work of nonconservative force for the systems (17).

4.3. Order of a Symmetric Scheme

Proposition 4. Consider the scheme (22) as mapping
𝜙Δ𝑡𝑞𝑘1,,𝑞𝑘𝑛,𝑝𝑘1,,𝑝𝑘𝑛=𝑞𝑘+11,,𝑞𝑘+1𝑛,𝑝𝑘+11,,𝑝𝑘+1𝑛,(31) and let 𝜙1Δ𝑡 be the inverse mapping of 𝜙Δ𝑡. Then, one has
𝜙1Δ𝑡=𝜙Δ𝑡.(32) That is, the scheme is symmetric

Proof. The inverse 𝜙1Δ𝑡 is obtained by exchanging (𝑝𝑘𝑠,𝑞𝑘𝑠) and (𝑝𝑘+1𝑠,𝑞𝑘+1𝑠). Replacing Δ𝑡 by Δ𝑡 and rearranging terms in 𝜙1Δ𝑡 leads to the mapping 𝜙1Δ𝑡. For this 𝜙1Δ𝑡, setting
𝑃𝑘+𝑚/𝑐𝑗𝑠𝑗=𝑃𝑘+1𝑚/𝑐𝑗𝑠𝑗,𝑠=1,2,,𝑛,𝑗=1,2,,𝑢,𝑚=0,1,,𝑐𝑗,(33) and omitting the tilde, we can obtain 𝜙Δ𝑡. Therefore, form (31) holds.

Proposition 5. If one chooses the weights 𝑑1,𝑑2,,𝑑𝑛 as (23), the accuracy of the scheme (22) is at least of order 2𝑛.

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 𝜙Δ𝑡 is 𝑂(Δ𝑡2𝑟+1) with a positive integer r. We first choose {𝑑𝑗}𝑢𝑗=1 such that
𝑢𝑗=1𝑑𝑗=1.(34) Since the error of 𝐼𝑚/𝑐𝑗,(𝑚1)/𝑐𝑗𝑃𝑠𝑗, 𝐼𝑚/𝑐𝑗,(𝑚1)/𝑐𝑗𝑄𝑠𝑗, and 𝑄𝑘𝑠𝑗 are 𝑂[(Δ𝑡/𝑐𝑗)3], the error of 𝜙Δ𝑡 is expressed as
𝑢𝑗=1𝑑𝑗×𝑐𝑗×𝑂Δ𝑡𝑐𝑗3=𝑢𝑗=1𝑐2𝑗𝑑𝑗𝑂Δ𝑡3.(35) If one chooses {𝑑𝑗}𝑢𝑗=1 such that
𝑛𝑗=1𝑐2𝑗𝑑𝑗=0,(36) then the 𝑂(Δ𝑡3)-term in the error of 𝜙Δ𝑡 vanishes. Since the error of 𝜙Δ𝑡 is of odd order, it becomes 𝑂(Δ𝑡5). The 𝑂(Δ𝑡5)-term in the error {𝑑𝑗}𝑢𝑗=1-term in the error of 𝜙Δ𝑡 is expressed as
𝑢𝑗=1𝑑𝑗×𝑐𝑗×𝑂Δ𝑡𝑐𝑗5=𝑢𝑗=1𝑐4𝑗𝑑𝑗𝑂Δ𝑡5.(37) These procedures can be repeated. The final condition for {𝑑𝑗}𝑢𝑗=1 is
𝑢𝑗=1𝑐2𝑢1𝑗𝑑𝑗=0.(38) Therefore, if one chooses {𝑑𝑗}𝑢𝑗=1 such that they satisfy the n simultaneous linear equations:
𝑢𝑗=1𝑐2𝑙𝑗𝑑𝑗=1for𝑙=0,0for𝑙=1,2,,𝑢1,(39) then the error of 𝜙Δ𝑡 is 𝑂(Δ𝑡2𝑢+1). Since the solution of (39) is given by (23), the order of accuracy is 2u.

5. A Numerical Example

Consider the motion of a particle with unit mass whose Hamiltonian is

𝐻𝑞,𝑝=12𝑝2,(40) and the motion of the system is subjected to nonpotential force

𝑄𝑠=cos𝑏𝑡,(41) where b is a constant.

The equation of motion of nonconservative particle is

𝑑𝑝𝑑𝑡=cos𝑏𝑡,𝑑𝑞𝑑𝑡=𝑝.(42) The analytic solution of (42) is given by

𝑝=𝐴𝑏sin𝑏𝑡,𝑞=𝐴𝑏2cos𝑏𝑡+𝜑,(43) which have the period 𝑇=2𝜋/𝑏. We take the initial conditions:

𝑝𝜋2𝑏=1,𝑞𝜋2=1,(44) and the calculation time 𝑡=𝑇. The parallel composition scheme with

𝑐𝑗=𝑗,𝑗=1,2,,𝑢(45) was used. We calculated the global error given by

𝑒𝑡=𝑝𝐾𝑝𝑇2+𝑞𝐾𝑞𝑇2,(46) 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(Δ𝑡2𝑛).

We should point out that the local error of the parallel composition is expressed as

𝑢𝑗=1𝑐2𝑢𝑗𝑑𝑗𝑂Δ𝑡2𝑛+1=1𝑐21𝑐22𝑐2𝑛𝑂Δ𝑡2𝑛+1,(47) the more the stages of the schemes are, the smaller the error of the scheme for nonconservative Hamiltonian systems.

6. 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.

Acknowledgments

This work is supported by the National Natural Science Foundation of China(10672143; 60575055) and the Natural Science Foundation of Henan Province, China (Grant no. 0511022200)