CONSERVATIVE METHODS FOR DYNAMICAL SYSTEMS

被引：16

作者：

Wan, Andy T. S. ^{[1
]}

Bihlo, Alexander ^{[2
]}

Nave, Jean-Christophe ^{[1
]}

机构：

[1] McGill Univ, Dept Math & Stat, Montreal, PQ H3A 0B9, Canada

[2] Mem Univ, Dept Math & Stat, St John, NF A1C 5S7, Canada

来源：

SIAM JOURNAL ON NUMERICAL ANALYSIS | 2017年 / 55卷 / 05期

基金：

加拿大自然科学与工程研究理事会;

关键词：

dynamical system; autonomous; nonautonomous; conserved quantity; first integral; finite difference; conservative methods; multiplier method; long-term stability; divided difference; RUNGE-KUTTA METHODS; DIFFERENTIAL-EQUATIONS; 1ST INTEGRALS; ENERGY; ODES; LAWS;

D O I：

10.1137/16M110719X

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We show a novel systematic way to construct conservative finite difference schemes for quasilinear first-order systems of ordinary differential equations with conserved quantities. In particular, this includes both autonomous and nonautonomous dynamical systems with conserved quantities of arbitrary forms, such as time-dependent conserved quantities. Sufficient conditions to construct conservative schemes of arbitrary order are derived using the multiplier method. General formulas for first-order conservative schemes are constructed using divided difference calculus. New conservative schemes are found for various dynamical systems such as Euler's equation of rigid body rotation, Lotka-Volterra systems, the planar restricted three-body problem, and the damped harmonic oscillator.

引用

页码：2255 / 2285

页数：31

共 25 条

[1]

Anco S.C., 2002, Symmetry and Integration Methods for Differential Equations

[2] Integrating factors and first integrals for ordinary differential equations [J].

Anco, SC ;

Bluman, G .

EUROPEAN JOURNAL OF APPLIED MATHEMATICS, 1998, 9 :245-259

[3] ON THE HAMILTONIAN INTERPOLATION OF NEAR-TO-THE-IDENTITY SYMPLECTIC MAPPINGS WITH APPLICATION TO SYMPLECTIC INTEGRATION ALGORITHMS [J].

BENETTIN, G ;

GIORGILLI, A .

JOURNAL OF STATISTICAL PHYSICS, 1994, 74 (5-6) :1117-1143

[4]

Bluman G., 2010, Applied Mathematical Sciences

[5] Numerical solution of isospectral flows [J].

Calvo, MP ;

Iserles, A ;

Zanna, A .

MATHEMATICS OF COMPUTATION, 1997, 66 (220) :1461-1486

[6]

Celledoni E., 2012, J PHYS A, V46

[7] ENERGY-PRESERVING RUNGE-KUTTA METHODS [J].

Celledoni, Elena ;

McLachlan, Robert I. ;

McLaren, David I. ;

Owren, Brynjulf ;

Quispel, G. Reinout W. ;

Wright, William M. .

ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE, 2009, 43 (04) :645-649

[8] STABILITY OF RUNGE-KUTTA METHODS FOR TRAJECTORY PROBLEMS [J].

COOPER, GJ .

IMA JOURNAL OF NUMERICAL ANALYSIS, 1987, 7 (01) :1-13

[9] Preserving multiple first integrals by discrete gradients [J].

Dahlby, Morten ;

Owren, Brynjulf ;

Yaguchi, Takaharu .

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, 2011, 44 (30)

[10]

Hairer E., 2006, GEOMETRIC NUMERICAL

← 1 2 3 →