Table of Contents
Research Letters in Physics
Volumeย 2008, Article IDย 168231, 5 pages
http://dx.doi.org/10.1155/2008/168231
Research Letter

An Energy-Work Relationship Integration Scheme for Nonconservative Hamiltonian Systems

1Institute of Mathematical Physics, Zhejiang Sci-Tech University, Hangzhou 310018, China
2College of Mechanics and Automatization Control, Zhejiang Sci-Tech University, Hangzhou 310018, China

Received 13 March 2008; Accepted 1 May 2008

Academic Editor: Eric G.ย Blackman

Copyright ยฉ 2008 Fu Jingli et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 ๐‘› multistage scheme of order 2 which its order of accuracy is 2๐‘›. 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 [1โ€“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 energy-conserving scheme is one of such geometric numerical integration scheme [4โ€“8]. 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) [1โ€“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 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 ๐‘ž๐‘ (๐‘ =1,โ€ฆ,๐‘›) and n gener-alized momentums ๐‘๐‘ (๐‘ =1,โ€ฆ,๐‘›). Suppose the system is subjected to n nonpotential generalized forces ๐‘„๎…ž๐‘ . The gener-alized Hamiltonian canonical equations of system as ฬ‡๐‘ž๐‘ =๐œ•๐ป๐œ•๐‘๐‘ ,ฬ‡๐‘๐‘ =โˆ’๐œ•๐ป๐œ•๐‘ž๐‘ +๐‘„๎…ž๐‘ ,๐‘ =1,2,โ€ฆ,๐‘›,(1) here, the Hamiltonian ๐ป=๐ป๎‚€๐‘ž1,โ€ฆ,๐‘ž๐‘›,๐‘1,โ€ฆ,๐‘๐‘›๎‚,(2) which represents the total energy. The relationship between the change of energy and the power of nonconservative force is easily verified as ๐‘‘๐ป๐‘‘๐‘ก=๐‘›๎“๐‘ =1๎‚€๐œ•๐ป๐œ•๐‘ž๐‘ ฬ‡๐‘ž๐‘ +๐œ•๐ป๐œ•๐‘๐‘ ฬ‡๐‘๐‘ ๎‚=๐‘›๎“๐‘ =1๎‚€๐œ•๐ป๐œ•๐‘ž๐‘ ๐œ•๐ป๐œ•๐‘๐‘ โˆ’๐œ•๐ป๐œ•๐‘๐‘ ๐œ•๐ป๐œ•๐‘ž๐‘ +๐œ•๐ป๐œ•๐‘๐‘ ๐‘„๎…ž๐‘ ๎‚=๐‘›๎“๐‘ =1๐œ•๐ป๐œ•๐‘๐‘ ๐‘„๎…ž๐‘ .(3) The numerical integration is considered as the discretization of ๐‘ž๐‘ ๎‚€๐‘ก๐‘˜+1๎‚=๐‘ž๐‘ ๎‚€๐‘ก๐‘˜๎‚+๎€œ๐‘ก๐‘˜+1๐‘ก๐‘˜๐œ•๐ป๎‚€๐‘ž1,โ€ฆ,๐‘ž๐‘›,๐‘1,โ€ฆ,๐‘๐‘›๎‚๐œ•๐‘๐‘ ๐‘‘๐‘ก,๐‘๐‘ ๎‚€๐‘ก๐‘˜+1๎‚=๐‘๐‘ ๎‚€๐‘ก๐‘˜๎‚โˆ’๎€œ๐‘ก๐‘˜+1๐‘ก๐‘˜๐œ•๐ป๎‚€๐‘ž1,โ€ฆ,๐‘ž๐‘›,๐‘1,โ€ฆ,๐‘๐‘›๎‚๐œ•๐‘ž๐‘ ๐‘‘๐‘ก+๐‘„๎…ž๐‘ ฮ”๐‘ก,๐‘ก๐‘˜=๐‘˜ฮ”๐‘ก,๐‘˜=0,1,2,โ€ฆ,(4) which are obtained by integrating both sides of (1) on the interval [๐‘ก๐‘˜,๐‘ก๐‘˜+1], where ฮ”๐‘ก is the step size.

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

3.1. Single-Stage Scheme

Let ๐‘๐‘˜๐‘  and ๐‘ž๐‘˜๐‘  be the numerical approximations of ๐‘๐‘ (๐‘ก๐‘˜) and ๐‘ž๐‘ (๐‘ก๐‘˜), respectively. Then a 1-stage scheme is given by

๐‘ž๐‘˜+1๐‘ =๐‘ž๐‘˜๐‘ +๐ผ1,0๐‘๐‘ ,๐‘๐‘˜+1๐‘ =๐‘๐‘˜๐‘ โˆ’๐ผ1,0๐‘ž๐‘ +๐‘„๎…ž๐‘ ฮ”๐‘ก,๐‘ =1,2,โ€ฆ,๐‘›,(5) with

๐ผ๐‘Ž,๐‘๐‘ž๐‘ =๎€ท๐‘Žโˆ’๐‘๎€ธฮ”๐‘ก๐›ฟ๐‘Ž,๐‘๐‘ž๐‘ ๐œ‡๐‘Ž,๐‘๐‘ž๐‘ ๐ป๐‘˜,๐ผ๐‘Ž,๐‘๐‘๐‘ =๎€ท๐‘Žโˆ’๐‘๎€ธฮ”๐‘ก๐›ฟ๐‘Ž,๐‘๐‘๐‘ ๐œ‡๐‘Ž,๐‘๐‘๐‘ ๐ป๐‘˜,๐ป๐‘˜=๐ป๎‚€๐‘ž๐‘˜1,โ€ฆ,๐‘ž๐‘˜๐‘›,๐‘๐‘˜1,โ€ฆ,๐‘๐‘˜๐‘›๎‚.(6) The notations ๐›ฟ๐‘Ž,๐‘๐‘๐‘  and ๐›ฟ๐‘Ž,๐‘๐‘ž๐‘  denote the partial difference quotient operators with respect to ๐‘๐‘  and ๐‘ž๐‘ , respectively, which are defined as

๐›ฟ๐‘Ž,๐‘๐‘ž๐‘ ๐น๎‚€๐‘ž๐‘™11,โ€ฆ,๐‘ž๐‘™๐‘›๐‘›,๐‘๐‘˜11,โ€ฆ,๐‘๐‘˜๐‘›๐‘›๎‚=๎‚€๐ธ๐‘Ž๐‘ž๐‘ โˆ’๐ธ๐‘๐‘ž๐‘ ๎‚๐น๎‚€๐ธ๐‘Ž๐‘ž๐‘ โˆ’๐ธ๐‘๐‘ž๐‘ ๎‚๐‘ž๐‘™๐‘ ๐‘ ,๐›ฟ๐‘Ž,๐‘๐‘๐‘ ๐น๎‚€๐‘ž๐‘™11,โ€ฆ,๐‘ž๐‘™๐‘›๐‘›,๐‘๐‘˜11,โ€ฆ,๐‘๐‘˜๐‘›๐‘›๎‚=๎‚€๐ธ๐‘Ž๐‘๐‘ โˆ’๐ธ๐‘๐‘๐‘ ๎‚๐น๎‚€๐ธ๐‘Ž๐‘๐‘ โˆ’๐ธ๐‘๐‘๐‘ ๎‚๐‘๐‘˜๐‘ ๐‘ ,(7) where ๐ธ๐‘Ž๐‘ž๐‘  and ๐ธ๐‘Ž๐‘๐‘  are the shift operators defined as

๐ธ๐‘Ž๐‘ž๐‘ ๐น๎‚€๐‘ž๐‘™11,โ€ฆ,๐‘ž๐‘™๐‘ โˆ’1๐‘ โˆ’1,๐‘ž๐‘™๐‘ ๐‘ ,๐‘ž๐‘™๐‘ +1๐‘ +1,โ€ฆ,๐‘ž๐‘™๐‘›๐‘›,๐‘๐‘˜11,โ€ฆ,๐‘๐‘˜๐‘›๐‘›๎‚=๐น๎‚€๐‘ž๐‘™11,โ€ฆ,๐‘ž๐‘™๐‘ โˆ’1๐‘ โˆ’1,๐‘ž๐‘™๐‘ +๐‘Ž๐‘ ,๐‘ž๐‘™๐‘ +1๐‘ +1,โ€ฆ,๐‘ž๐‘™๐‘›๐‘›,๐‘๐‘˜11,โ€ฆ,๐‘๐‘˜๐‘›๐‘›๎‚,๐ธ๐‘Ž๐‘๐‘ ๐น๎‚€๐‘ž๐‘™11,โ€ฆ,๐‘ž๐‘™๐‘›๐‘›,๐‘๐‘˜11,โ€ฆ,๐‘๐‘˜๐‘ โˆ’1๐‘ โˆ’1,๐‘๐‘˜๐‘ ๐‘ ,๐‘๐‘˜๐‘ +1๐‘ +1,โ€ฆ,๐‘๐‘˜๐‘›๐‘›๎‚=๐น๎‚€๐‘ž๐‘™11,โ€ฆ,๐‘ž๐‘™๐‘›๐‘›,๐‘๐‘˜11,โ€ฆ,๐‘๐‘˜๐‘ โˆ’1๐‘ โˆ’1,๐‘๐‘˜๐‘ +๐‘Ž๐‘ ,๐‘๐‘˜๐‘ +1๐‘ +1,โ€ฆ,๐‘๐‘˜๐‘›๐‘›๎‚.(8) The notations ๐œ‡๐‘Ž,๐‘๐‘ž๐‘  and ๐œ‡๐‘Ž,๐‘๐‘๐‘  denote the mean difference operators with respect to all variables except for ๐‘ž๐‘  and ๐‘๐‘ , respectively, which are defined as

๐œ‡๐‘Ž,๐‘๐‘ž๐‘ ๐น๎‚€๐‘ž๐‘™11,โ€ฆ,๐‘ž๐‘™๐‘›๐‘›,๐‘๐‘˜11,โ€ฆ,๐‘๐‘˜๐‘›๐‘›๎‚=๐‘€๐‘Ž,๐‘๎‚€๐ธ๐‘ž1,โ€ฆ,๐ธ๐‘ž๐‘ โˆ’1,๐ธ๐‘ž๐‘ +1,โ€ฆ,๐ธ๐‘ž๐‘›,๐ธ๐‘1,โ€ฆ,๐ธ๐‘๐‘›๎‚,๐œ‡๐‘Ž,๐‘๐‘ž๐‘ ๐น๎‚€๐‘ž๐‘™11,โ€ฆ,๐‘ž๐‘™๐‘›๐‘›,๐‘๐‘˜11,โ€ฆ,๐‘๐‘˜๐‘›๐‘›๎‚=๐‘€๐‘Ž,๐‘๎‚€๐ธ๐‘ž1,โ€ฆ,๐ธ๐‘ž๐‘›,๐ธ๐‘1,โ€ฆ,๐ธ๐‘๐‘ โˆ’1,๐ธ๐‘๐‘ +1,โ€ฆ,๐ธ๐‘๐‘›๎‚,(9) with

๐‘€๐‘Ž,๐‘๎‚€๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘Ÿโˆ’1๎‚=1๐‘Ÿ!๐‘Ÿ๎“๐‘™=1perโŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ๐‘ฅ๐‘Ž1๐‘ฅ๐‘Ž2โ‹ฏ๐‘ฅ๐‘Ž๐‘Ÿโˆ’1โ‹ฎโ‹ฎโ‹ฎ๐‘ฅ๐‘Ž1๐‘ฅ๐‘Ž2โ‹ฏ๐‘ฅ๐‘Ž๐‘Ÿโˆ’1๐‘ฅ๐‘1๐‘ฅ๐‘2โ‹ฏ๐‘ฅ๐‘๐‘Ÿโˆ’1โ‹ฎโ‹ฎโ‹ฎ๐‘ฅ๐‘1๐‘ฅ๐‘2โ‹ฏ๐‘ฅ๐‘๐‘Ÿโˆ’1โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ 25๐‘๐‘ก}๐‘Ÿโˆ’125๐‘๐‘ก}๐‘™โˆ’1,(10) where per(A) denotes the permanent or plus determinant of a matrix A [10]. For example, in the case d = 1, we have

๐œ‡๐‘Ž,๐‘๐‘ž1=๐‘€๐‘Ž,๐‘๎‚€๐ธ๐‘ž1๎‚=12๎‚€๐ธ๐‘Ž๐‘ž1+๐ธ๐‘๐‘ž1๎‚,๐œ‡๐‘Ž,๐‘๐‘1=๐‘€๐‘Ž,๐‘๎‚€๐ธ๐‘1๎‚=12๎‚€๐ธ๐‘Ž๐‘1+๐ธ๐‘๐‘1๎‚.(11) The operators ๐›ฟ๐‘Ž,๐‘๐‘ž๐‘ ,๐›ฟ๐‘Ž,๐‘๐‘๐‘ ,๐œ‡๐‘Ž,๐‘๐‘ž๐‘ , and ๐œ‡๐‘Ž,๐‘๐‘๐‘  have symmetry expressed as

๐›ฟ๐‘Ž,๐‘๐‘ž๐‘ =๐›ฟ๐‘,๐‘Ž๐‘ž๐‘ ,๐›ฟ๐‘Ž,๐‘๐‘๐‘ =๐›ฟ๐‘,๐‘Ž๐‘๐‘ ,๐œ‡๐‘Ž,๐‘๐‘ž๐‘ =๐œ‡๐‘,๐‘Ž๐‘ž๐‘ ,๐œ‡๐‘Ž,๐‘๐‘๐‘ =๐œ‡๐‘,๐‘Ž๐‘๐‘ .(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:
๐ป๐‘˜+๐‘Žโˆ’๐ป๐‘˜+๐‘=๐‘›๎“๐‘ =1๎‚ƒ๎‚€๐›ฟ๐‘Ž,๐‘๐‘ž๐‘ ๐œ‡๐‘Ž,๐‘๐‘ž๐‘ ๐ป๐‘˜๎‚๎‚€๐‘ž๐‘˜+๐‘Ž๐‘ โˆ’๐‘ž๐‘˜+๐‘๐‘ ๎‚+๎‚€๐›ฟ๐‘Ž,๐‘๐‘๐‘ ๐œ‡๐‘Ž,๐‘๐‘๐‘ ๐ป๐‘˜๎‚๎‚€๐‘๐‘˜+๐‘Ž๐‘ โˆ’๐‘๐‘˜+๐‘๐‘ ๎‚๎‚„.(13)

Proof. For simplicity, we set
๐ธ๐‘Ž=๎‚€๐ธ๐‘Ž๐‘ž1,โ€ฆ,๐ธ๐‘Ž๐‘ž๐‘›,๐ธ๐‘Ž๐‘1,โ€ฆ,๐ธ๐‘Ž๐‘๐‘›๎‚๐‘‡.(14) We first note the identity
๐‘›๎‘๐‘ ๐ธ๐‘Ž๐‘ž๐‘ ๐ธ๐‘Ž๐‘๐‘ =1๎€ท2๐‘›๎€ธ!per(โ†’๐ธ๐‘Ž,โ€ฆ,โ†’๐ธ๐‘Ž๎„ฟ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…ƒ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…Œ2๐‘›).(15) It follows that ๐ป๐‘˜+๐‘Žโˆ’๐ป๐‘˜+๐‘=(๐‘›๎‘๐‘ =1๐ธ๐‘Ž๐‘ž๐‘ ๐ธ๐‘Ž๐‘๐‘ โˆ’๐‘›๎‘๐‘ =1๐ธ๐‘๐‘ž๐‘ ๐ธ๐‘๐‘๐‘ )๐ป๐‘˜=๐‘›๎“๐‘ =1๎‚ƒ๎‚€๐ธ๐‘Ž๐‘ž๐‘ โˆ’๐ธ๐‘๐‘ž๐‘ ๎‚๐‘€๐‘Ž,๐‘๎‚€๐ธ๐‘ž1,โ€ฆ,๐ธ๐‘ž๐‘›,๐ธ๐‘1,โ€ฆ,๐ธ๐‘๐‘ โˆ’1,๐ธ๐‘๐‘ +1,โ€ฆ,๐ธ๐‘๐‘›๎‚+๎‚€๐ธ๐‘Ž๐‘๐‘ โˆ’๐ธ๐‘๐‘๐‘ ๎‚๐‘€๐‘Ž,๐‘๎‚€๐ธ๐‘ž1,โ€ฆ,๐ธ๐‘ž๐‘ โˆ’1,๐ธ๐‘ž๐‘ +1,โ€ฆ,๐ธ๐‘ž๐‘›,๐ธ๐‘1,โ€ฆ,๐ธ๐‘๐‘›๎‚๎‚„๐ป๐‘˜=๐‘›๎“๐‘ =1๎‚ƒ๎‚€๐›ฟ๐‘Ž,๐‘๐‘ž๐‘ ๐œ‡๐‘Ž,๐‘๐‘ž๐‘ ๐ป๐‘˜๎‚๎‚€๐‘ž๐‘˜+๐‘Ž๐‘ โˆ’๐‘ž๐‘˜+๐‘๐‘ ๎‚+๎‚€๐›ฟ๐‘Ž,๐‘๐‘๐‘ ๐œ‡๐‘Ž,๐‘๐‘๐‘ ๐ป๐‘˜๎‚๎‚€๐‘๐‘˜+๐‘Ž๐‘ โˆ’๐‘๐‘˜+๐‘๐‘ ๎‚๎‚„,(16) where we have used the properties of the permanent and the definitions of operators (7)โ€“(10) [10].

Proposition 2. The scheme {(1),โ€ฆ,(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 ๐ป๐‘˜ is equivalent to the work of nonconservative force:
๐ป๐‘˜+1โˆ’๐ป๐‘˜=๐‘›๎“๐‘ =1๎‚ƒ๎‚€๐›ฟ1,0๐‘ž๐‘ ๐œ‡1,0๐‘ž๐‘ ๐ป๐‘˜๎‚๎‚€๐‘ž๐‘˜+1๐‘ โˆ’๐‘ž๐‘˜๐‘ ๎‚+๎‚€๐›ฟ1,0๐‘๐‘ ๐œ‡1,0๐‘๐‘ ๐ป๐‘˜๎‚๎‚€๐‘๐‘˜+1๐‘ โˆ’๐‘๐‘˜๐‘ ๎‚๎‚„=ฮ”๐‘ก๐‘›๎“๐‘ =1๎‚ƒ๎‚€๐›ฟ1,0๐‘ž๐‘ ๐œ‡1,0๐‘ž๐‘ ๐ป๐‘˜๎‚๎‚€๐›ฟ1,0๐‘๐‘ ๐œ‡1,0๐‘๐‘ ๐ป๐‘˜๎‚+๎‚€๐›ฟ1,0๐‘๐‘ ๐œ‡1,0๐‘๐‘ ๐ป๐‘˜๎‚๎‚€โˆ’๐›ฟ1,0๐‘ž๐‘ ๐œ‡1,0๐‘ž๐‘ ๐ป๐‘˜+๐‘„๎…ž๐‘ ๎‚๎‚„=๐‘›๎“๐‘ =1ฮ”๐‘ก๎‚€๐›ฟ1,0๐‘๐‘ ๐œ‡1,0๐‘๐‘ ๐ป๐‘˜๎‚๐‘„๎…ž๐‘ ,(17) which is a discrete analog of (3).

3.3. Order of Accuracy

The local errors involved in the determination of {๐‘๐‘˜+1๐‘ ,๐‘ž๐‘˜+1๐‘ }๐‘›๐‘ =1 from {๐‘๐‘˜๐‘ ,๐‘ž๐‘˜๐‘ }๐‘›๐‘ =1 are ๐‘‚(ฮ”๐‘ก3), that is, ๐ผ1,0๐‘ž๐‘  and ๐ผ1,0๐‘๐‘  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 ฮ”๐‘ก/๐‘ in series:

๐‘ƒ๐‘˜+๐‘š/๐‘๐‘ =๐‘ƒ๐‘˜+(๐‘šโˆ’1)/๐‘๐‘ โˆ’๐ผ๐‘š/๐‘,(๐‘šโˆ’1)/๐‘๐‘„๐‘ +๐‘„๎…ž๐‘ ๎‚€๐‘„๐‘˜1,โ€ฆ,๐‘„๐‘˜๐‘›,๐‘ƒ๐‘˜1,โ€ฆ,๐‘ƒ๐‘˜๐‘›๎‚ฮ”๐‘ก๐‘,๐‘„๐‘˜+๐‘š/๐‘๐‘ =๐‘„๐‘˜+(๐‘šโˆ’1)/๐‘๐‘ +๐ผ๐‘š/๐‘,(๐‘šโˆ’1)/๐‘๐‘ƒ๐‘ ๐‘ƒ๐‘˜+1๐‘ ๐‘๐‘˜+1๐‘ ,๐‘ƒ๐‘˜๐‘ =๐‘๐‘˜๐‘ ,๐‘„๐‘˜+1๐‘ =๐‘ž๐‘˜+1๐‘ ,๐‘„๐‘˜๐‘ =๐‘ž๐‘˜๐‘ ,๐‘ =1,2,โ€ฆ,๐‘›,๐‘š=1,2,โ€ฆ,๐‘,(18) with

๐ผ๐‘Ž,๐‘๐‘„๐‘ =๎€ท๐‘Žโˆ’๐‘๎€ธฮ”๐‘ก๐›ฟ๐‘Ž,๐‘๐‘„๐‘ ๐œ‡๐‘Ž,๐‘๐‘„๐‘ ๐ป๎‚€๐‘„๐‘˜1,โ€ฆ,๐‘„๐‘˜๐‘›,๐‘ƒ๐‘˜1,โ€ฆ,๐‘ƒ๐‘˜๐‘›๎‚,๐ผ๐‘Ž,๐‘๐‘ƒ๐‘ =๎€ท๐‘Žโˆ’๐‘๎€ธฮ”๐‘ก๐›ฟ๐‘Ž,๐‘๐‘ƒ๐‘ ๐œ‡๐‘Ž,๐‘๐‘ƒ๐‘ ๐ป๎‚€๐‘„๐‘˜1,โ€ฆ,๐‘„๐‘˜๐‘›,๐‘ƒ๐‘˜1,โ€ฆ,๐‘ƒ๐‘˜๐‘›๎‚,(19) where ๐‘ƒ๐‘˜+๐‘š/๐‘๐‘  and ๐‘„๐‘˜+๐‘š/๐‘๐‘  are the internal stage variables. It should be noted that the above scheme is equivalent to the scheme:

๐‘๐‘˜+1๐‘ =๐‘๐‘˜๐‘ โˆ’๐‘๎“๐‘™=1๐ผ๐‘™/๐‘,(๐‘™โˆ’1)/๐‘๐‘„๐‘ +๐‘„๎…ž๐‘ ๎‚€๐‘ž๐‘˜1,โ€ฆ,๐‘ž๐‘˜๐‘›,๐‘๐‘˜1,โ€ฆ,๐‘๐‘˜๐‘›๎‚ฮ”๐‘ก๐‘,๐‘ž๐‘˜+1๐‘ =๐‘ž๐‘˜๐‘ โˆ’๐‘๎“๐‘™=1๐ผ๐‘™/๐‘,(๐‘™โˆ’1)/๐‘๐‘ƒ๐‘ ,๐‘ƒ๐‘˜+๐‘š/๐‘๐‘ =๐‘โˆ’๐‘š๐‘(๐‘๐‘˜๐‘ โˆ’๐‘š๎“๐‘™=1๐ผ๐‘™/๐‘,(๐‘™โˆ’1)/๐‘๐‘„๐‘ +๐‘„๎…ž๐‘ ฮ”๐‘ก๐‘)+๐‘š๐‘(๐‘๐‘˜+1๐‘ +๐‘๎“๐‘™=๐‘š+1๐ผ๐‘™/๐‘,(๐‘™โˆ’1)/๐‘๐‘„๐‘ +๐‘„๎…ž๐‘ ฮ”๐‘ก๐‘),๐‘„๐‘˜+๐‘š/๐‘๐‘ =๐‘โˆ’๐‘š๐‘(๐‘ž๐‘˜๐‘ +๐‘š๎“๐‘™=1๐ผ๐‘™/๐‘,(๐‘™โˆ’1)/๐‘๐‘ƒ๐‘ )+๐‘š๐‘(๐‘ž๐‘˜+1๐‘ โˆ’๐‘๎“๐‘™=๐‘š+1๐ผ๐‘™/๐‘,(๐‘™โˆ’1)/๐‘๐‘ƒ๐‘ ),๐‘ =1,2,โ€ฆ,๐‘›,๐‘š=1,2,โ€ฆ,๐‘โˆ’1.(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 ๐‘ร—๐‘‚[(ฮ”๐‘ก/๐‘)3]=๐‘โˆ’2๐‘‚(ฮ”๐‘ก)3.

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

4.1. Parallel Composition Scheme

Let ๐‘1,๐‘2,โ€ฆ,๐‘๐‘› be arbitrary positive integers satisfying

๐‘1<๐‘2<โ‹ฏ<๐‘๐‘›,(21) then a new scheme is constructed by connection ๐‘1-stage,๐‘2-stage,โ€ฆ,๐‘๐‘›-stage schemes of order 2 in parallel: ๐‘๐‘˜+1๐‘ =๐‘๐‘˜๐‘ โˆ’๐‘ข๎“๐‘—=1๐‘‘๐‘—๐‘๐‘—๎“๐‘™=1๐ผ๐‘™/๐‘๐‘—,(๐‘™โˆ’1)/๐‘๐‘—๐‘„๐‘ ๐‘—+๐‘„๎…ž๐‘ ๎‚€๐‘ž๐‘˜1,โ€ฆ,๐‘ž๐‘˜๐‘›,๐‘๐‘˜1,โ€ฆ,๐‘๐‘˜๐‘›๎‚ฮ”๐‘ก,๐‘ž๐‘˜+1๐‘ =๐‘ž๐‘˜๐‘ +๐‘ข๎“๐‘—=1๐‘‘๐‘—๐‘๐‘—๎“๐‘™=1๐ผ๐‘™/๐‘๐‘—,(๐‘™โˆ’1)/๐‘๐‘—๐‘ƒ๐‘–๐‘—,๐‘ƒ๐‘˜+๐‘š/๐‘๐‘—๐‘ ๐‘—=๐‘๐‘—โˆ’๐‘š๐‘๐‘—(๐‘๐‘˜๐‘ โˆ’๐‘š๎“๐‘™=1๐ผ๐‘™/๐‘๐‘—,(๐‘™โˆ’1)/๐‘๐‘—๐‘„๐‘ ๐‘—+๐‘„๎…ž๐‘ ๐‘—ฮ”๐‘ก)+๐‘š๐‘๐‘—(๐‘๐‘˜+1๐‘ โˆ’๐‘š๎“๐‘™=๐‘š+1๐ผ๐‘™/๐‘๐‘—,(๐‘™โˆ’1)/๐‘๐‘—๐‘„๐‘ ๐‘—+๐‘„๎…ž๐‘ ๐‘—ฮ”๐‘ก),๐‘„๐‘˜+๐‘š/๐‘๐‘—๐‘ ๐‘—=๐‘๐‘—โˆ’๐‘š๐‘๐‘—(๐‘ž๐‘˜๐‘ +๐‘š๎“๐‘™=1๐ผ๐‘™/๐‘๐‘—,(๐‘™โˆ’1)/๐‘๐‘—๐‘„๐‘ ๐‘—)+๐‘š๐‘๐‘—(๐‘ž๐‘˜+1๐‘ +๐‘š๎“๐‘™=๐‘š+1๐ผ๐‘™/๐‘๐‘—,(๐‘™โˆ’1)/๐‘๐‘—๐‘„๐‘ ๐‘—),๐‘ƒ๐‘˜+1๐‘ ๐‘—=๐‘๐‘˜+1๐‘ ,๐‘ƒ๐‘˜๐‘ ๐‘—=๐‘๐‘˜๐‘ ,๐‘„๐‘˜+1๐‘ ๐‘—=๐‘ž๐‘˜+1๐‘ ,๐‘„๐‘˜๐‘ ๐‘—=๐‘ž๐‘˜๐‘ ,๐‘ =1,2,โ€ฆ,๐‘›,๐‘—=1,2,โ€ฆ,๐‘ข,๐‘š=1,2,โ€ฆ,๐‘๐‘—โˆ’1,(22) with the weights

๐‘‘๐‘—=โŽงโŽชโŽชโŽจโŽชโŽชโŽฉ1for๐‘ข=1,๐‘2๐‘ขโˆ’2๐‘—โˆ๐‘ข๐‘™=1,๐‘™โ‰ ๐‘—๎‚€๐‘2๐‘—โˆ’๐‘2๐‘™๎‚for๐‘ขโ‰ฅ2,๐‘—=1,2,โ€ฆ,๐‘ข,(23) where

๐ผ๐‘Ž,๐‘๐‘ƒ๐‘ ๐‘—=๎€ท๐‘Žโˆ’๐‘๎€ธฮ”๐‘ก๐›ฟ๐‘Ž,๐‘๐‘ƒ๐‘ ๐‘—๐œ‡๐‘Ž,๐‘๐‘ƒ๐‘ ๐‘—๐ป๐‘˜๐‘—,๐ผ๐‘Ž,๐‘๐‘„๐‘ ๐‘—=๎€ท๐‘Žโˆ’๐‘๎€ธฮ”๐‘ก๐›ฟ๐‘Ž,๐‘๐‘„๐‘ ๐‘—๐œ‡๐‘Ž,๐‘๐‘„๐‘ ๐‘—๐ป๐‘˜๐‘—,๐ป๐‘˜๐‘—=๐ป๎‚€๐‘„๐‘˜1๐‘—,โ€ฆ,๐‘„๐‘˜๐‘›๐‘—,๐‘ƒ๐‘˜1๐‘—,โ€ฆ,๐‘ƒ๐‘˜๐‘›๐‘—๎‚,ฮ›๐‘˜๐‘ ๐‘—=ฮ›๐‘ ๎‚€๐‘„๐‘˜1๐‘—,โ€ฆ,๐‘„๐‘˜๐‘›๐‘—,๐‘ƒ๐‘˜1๐‘—,โ€ฆ,๐‘ƒ๐‘˜๐‘›๐‘—๎‚.(24)

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

Proposition 3. The scheme (23) with the condition
๐‘ข๎“๐‘—=1๐‘‘๐‘—=1(25) satisfies relationship between the change of energy and work of nonconservative force for nonconservative Hamiltonian systems

Proof. We first note
๐ป๐‘˜+1=๐ป๐‘˜+1๐‘—,๐ป๐‘˜=๐ป๐‘˜๐‘—,๐‘—=1,2,โ€ฆ,๐‘ข.(26) We see from Proposition 1 that
๐ป๐‘˜+๐‘Ž๐‘—โˆ’๐ป๐‘˜+๐‘๐‘—=๐‘›๎“๐‘ =1๎‚€๐›ฟ๐‘Ž,๐‘๐‘ƒ๐‘ ๐‘—๐œ‡๐‘Ž,๐‘๐‘ƒ๐‘ ๐‘—๐ป๐‘˜๐‘—๎‚๎‚€๐‘ƒ๐‘˜+๐‘Ž๐‘ ๐‘—โˆ’๐‘ƒ๐‘˜+๐‘๐‘ ๐‘—๎‚+๎‚€๐›ฟ๐‘Ž,๐‘๐‘„๐‘ ๐‘—๐œ‡๐‘Ž,๐‘๐‘„๐‘ ๐‘—๐ป๐‘˜๐‘—๎‚๎‚€๐‘„๐‘˜+๐‘Ž๐‘ ๐‘—โˆ’๐‘„๐‘˜+๐‘๐‘ ๐‘—๎‚.(27) It follows from (25)โ€“(27) that
๐ป๐‘˜+1โˆ’๐ป๐‘˜=๐‘ข๎“๐‘—=1๐‘‘๐‘—๎‚€๐ป๐‘˜+1๐‘—โˆ’๐ป๐‘˜๐‘—๎‚=๐‘ข๎“๐‘—=1๐‘‘๐‘—๐‘๐‘—๎“๐‘š=1๎‚€๐ป๐‘˜+๐‘š/๐‘๐‘—๐‘—โˆ’๐ป๐‘˜+(๐‘šโˆ’1)/๐‘๐‘—๐‘—๎‚=๐‘ข๎“๐‘—=1๐‘‘๐‘—๐‘๐‘—ฮ”๐‘ก๐‘๐‘—๎“๐‘š=1๐‘›๎“๐‘ =1๎‚ƒ๐ผ๐‘˜+๐‘š/๐‘๐‘—,๐‘˜+(๐‘šโˆ’1)/๐‘๐‘—๐‘ƒ๐‘ ๐‘—ร—๎‚€๐‘ƒ๐‘˜+๐‘š/๐‘๐‘—๐‘ ๐‘—โˆ’๐‘ƒ๐‘˜+(๐‘šโˆ’1)/๐‘๐‘—๐‘ ๐‘—๎‚+๐ผ๐‘˜+๐‘š/๐‘๐‘—,๐‘˜+(๐‘šโˆ’1)/๐‘๐‘—๐‘„๐‘ ๐‘—ร—๎‚€๐‘„๐‘˜+๐‘š/๐‘๐‘—๐‘ ๐‘—โˆ’๐‘„๐‘˜+(๐‘šโˆ’1)/๐‘๐‘—๐‘ ๐‘—๎‚๎‚„.(28) We obtain from (22)
๐‘ƒ๐‘˜+๐‘š/๐‘๐‘—๐‘ ๐‘—โˆ’๐‘ƒ๐‘˜+(๐‘šโˆ’1)/๐‘๐‘—๐‘ ๐‘—=1๐‘๐‘—(๐‘ข๎“๐‘Ÿ=1๐‘๐‘Ÿ๐‘๐‘—๎“๐‘™=1๐ผ๐‘™/๐‘๐‘Ÿ,(๐‘™โˆ’1)/๐‘๐‘Ÿ๐‘ƒ๐‘ ๐‘Ÿโˆ’๐‘๐‘—๎“๐‘™=1๐ผ๐‘™/๐‘๐‘—,(๐‘™โˆ’1)/๐‘๐‘—๐‘ƒ๐‘ ๐‘—)+๐ผ๐‘š/๐‘๐‘—,(๐‘šโˆ’1)/๐‘๐‘—๐‘ƒ๐‘ ๐‘—(29) Substituting (29) into (28) yields
๐ป๐‘˜+1โˆ’๐ป๐‘˜=๐‘ข๎“๐‘—=1๐‘‘๐‘—๐‘๐‘—๎“๐‘š=1๐‘๐‘—๐‘›๎“๐‘ =1๐ผ๐‘š/๐‘๐‘—,(๐‘šโˆ’1)/๐‘๐‘—๐‘ƒ๐‘ ๐‘—๐‘„๎…ž๐‘ ๐‘—,(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
๐œ™ฮ”๐‘กโˆถ๎‚€๐‘ž๐‘˜1,โ€ฆ,๐‘ž๐‘˜๐‘›,๐‘๐‘˜1,โ€ฆ,๐‘๐‘˜๐‘›๎‚=๎‚€๐‘ž๐‘˜+11,โ€ฆ,๐‘ž๐‘˜+1๐‘›,๐‘๐‘˜+11,โ€ฆ,๐‘๐‘˜+1๐‘›๎‚,(31) and let ๐œ™โˆ’1ฮ”๐‘ก be the inverse mapping of ๐œ™ฮ”๐‘ก. Then, one has
๐œ™โˆ’1โˆ’ฮ”๐‘ก=๐œ™ฮ”๐‘ก.(32) That is, the scheme is symmetric

Proof. The inverse ๐œ™โˆ’1ฮ”๐‘ก is obtained by exchanging (๐‘๐‘˜๐‘ ,๐‘ž๐‘˜๐‘ ) and (๐‘๐‘˜+1๐‘ ,๐‘ž๐‘˜+1๐‘ ). Replacing ฮ”๐‘ก by โˆ’ฮ”๐‘ก and rearranging terms in ๐œ™โˆ’1ฮ”๐‘ก leads to the mapping ๐œ™โˆ’1โˆ’ฮ”๐‘ก. For this ๐œ™โˆ’1โˆ’ฮ”๐‘ก, setting
๐‘ƒ๐‘˜+๐‘š/๐‘๐‘—๐‘ ๐‘—=๐‘ƒ๐‘˜+1โˆ’๐‘š/๐‘๐‘—๐‘ ๐‘—,๐‘ =1,2,โ€ฆ,๐‘›,๐‘—=1,2,โ€ฆ,๐‘ข,๐‘š=0,1,โ€ฆ,๐‘๐‘—,(33) and omitting the tilde, we can obtain ๐œ™ฮ”๐‘ก. Therefore, form (31) holds.

Proposition 5. If one chooses the weights ๐‘‘1,๐‘‘2,โ€ฆ,๐‘‘๐‘› as (23), the accuracy of the scheme (22) is at least of order 2๐‘›.

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 ๐œ™ฮ”๐‘ก is ๐‘‚(ฮ”๐‘ก2๐‘Ÿ+1) with a positive integer r. We first choose {๐‘‘๐‘—}๐‘ข๐‘—=1 such that
๐‘ข๎“๐‘—=1๐‘‘๐‘—=1.(34) Since the error of ๐ผ๐‘š/๐‘๐‘—,(๐‘šโˆ’1)/๐‘๐‘—๐‘ƒ๐‘ ๐‘—, ๐ผ๐‘š/๐‘๐‘—,(๐‘šโˆ’1)/๐‘๐‘—๐‘„๐‘ ๐‘—, and ๐‘„๎…ž๐‘˜๐‘ ๐‘— are ๐‘‚[(ฮ”๐‘ก/๐‘๐‘—)3], the error of ๐œ™ฮ”๐‘ก is expressed as
๐‘ข๎“๐‘—=1๐‘‘๐‘—ร—๐‘๐‘—ร—๐‘‚๎‚ƒ๎‚€ฮ”๐‘ก๐‘๐‘—๎‚3๎‚„=๐‘ข๎“๐‘—=1๐‘โˆ’2๐‘—๐‘‘๐‘—๐‘‚๎‚€ฮ”๐‘ก3๎‚.(35) If one chooses {๐‘‘๐‘—}๐‘ข๐‘—=1 such that
๐‘›๎“๐‘—=1๐‘โˆ’2๐‘—๐‘‘๐‘—=0,(36) then the ๐‘‚(ฮ”๐‘ก3)-term in the error of ๐œ™ฮ”๐‘ก vanishes. Since the error of ๐œ™ฮ”๐‘ก is of odd order, it becomes ๐‘‚(ฮ”๐‘ก5). The ๐‘‚(ฮ”๐‘ก5)-term in the error {๐‘‘๐‘—}๐‘ข๐‘—=1-term in the error of ๐œ™ฮ”๐‘ก is expressed as
๐‘ข๎“๐‘—=1๐‘‘๐‘—ร—๐‘๐‘—ร—๐‘‚๎‚ƒ๎‚€ฮ”๐‘ก๐‘๐‘—๎‚5๎‚„=๐‘ข๎“๐‘—=1๐‘โˆ’4๐‘—๐‘‘๐‘—๐‘‚๎‚€ฮ”๐‘ก5๎‚.(37) These procedures can be repeated. The final condition for {๐‘‘๐‘—}๐‘ข๐‘—=1 is
๐‘ข๎“๐‘—=1๐‘โˆ’2๎€ท๐‘ขโˆ’1๎€ธ๐‘—๐‘‘๐‘—=0.(38) Therefore, if one chooses {๐‘‘๐‘—}๐‘ข๐‘—=1 such that they satisfy the n simultaneous linear equations:
๐‘ข๎“๐‘—=1๐‘โˆ’2๐‘™๐‘—๐‘‘๐‘—=โŽงโŽชโŽจโŽชโŽฉ1for๐‘™=0,0for๐‘™=1,2,โ€ฆ,๐‘ขโˆ’1,(39) then the error of ๐œ™ฮ”๐‘ก is ๐‘‚(ฮ”๐‘ก2๐‘ข+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

๐ป๎€ท๐‘ž,๐‘๎€ธ=12๐‘2,(40) and the motion of the system is subjected to nonpotential force

๐‘„๎…ž๐‘ =cos๐‘๐‘ก,(41) where b is a constant.

The equation of motion of nonconservative particle is

๐‘‘๐‘๐‘‘๐‘ก=cos๐‘๐‘ก,๐‘‘๐‘ž๐‘‘๐‘ก=๐‘.(42) The analytic solution of (42) is given by

๐‘=๐ด๐‘sin๐‘๐‘ก,๐‘ž=โˆ’๐ด๐‘2cos๎€ท๐‘๐‘ก+๐œ‘๎€ธ,(43) which have the period ๐‘‡=2๐œ‹/๐‘. We take the initial conditions:

๐‘๎‚€๐œ‹2๐‘๎‚=1,๐‘ž๎‚€๐œ‹2๎‚=1,(44) and the calculation time ๐‘ก=๐‘‡. The parallel composition scheme with

๐‘๐‘—=๐‘—,๐‘—=1,2,โ€ฆ,๐‘ข(45) was used. We calculated the global error given by

๐‘’๎€ท๐‘ก๎€ธ=๎‚™๎‚€๐‘๐พโˆ’๐‘๎€ท๐‘‡๎€ธ๎‚2+๎‚€๐‘ž๐พโˆ’๐‘ž๎€ท๐‘‡๎€ธ๎‚2,(46) where ๐พ=๐‘‡/ฮ”๐‘ก. Since the global error e(t) is about ๐‘‡/ฮ”๐‘ก times the local error, e(t) is expressed as ๐‘’(๐‘‡)=๐‘‚(ฮ”๐‘ก2๐‘›).

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

๐‘ข๎“๐‘—=1๐‘โˆ’2๐‘ข๐‘—๐‘‘๐‘—๐‘‚๎‚€ฮ”๐‘ก2๐‘›+1๎‚=1๐‘21๐‘22โ‹ฏ๐‘2๐‘›๐‘‚๎‚€ฮ”๐‘ก2๐‘›+1๎‚,(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)

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