In this paper, explicit Runge–Kutta methods are investigated for numerical solutions of nonlinear dynamical systems with conserved quantities. The concept, ε-preserving is introduced to describe the conserved quantities being approximately retained. Then, a modified version of explicit Runge–Kutta methods based on the optimization technique is presented. With respect to the computational effort, the modified Runge–Kutta method is superior to implicit numerical methods in the literature. The order of the modified Runge–Kutta method is the same as the standard Runge–Kutta method, but it is superior in preserving the conserved quantities to the standard one. Numerical experiments are provided to illustrate the effectiveness of the modified Runge–Kutta method.
Issue Section:
Research Papers
References
1.
Leimkuhler
, B.
, and Reich
, S.
, 2004
, Simulating Hamiltonian Dynamics
, Cambridge University Press
, Cambridge, UK
.2.
Greiner
, W.
, 2010
, Classical Mechanics: Systems of Particles and Hamiltonian Dynamics
, 2nd ed., Springer-Verlag
, Berlin
.3.
Lee
, T.
, Leok
, M.
, and McClamroch
, N. H.
, 2007
, “Lie Variational Integrators for the Full Body Problem in Orbital Mechanics
,” Celestial Mech. Dyn. Astron.
, 98
(2
), pp. 121
–144
.4.
Li
, S.
, 2007
, “Introduction to Classical Mechanics
,” University of Science and Technology of China, Hefei, China (in Chinese).5.
Griffiths
, D. F.
, and Higham
, D. J.
, 2010
, Numerical Methods for Ordinary Differential Equations
, Springer-Verlag
, London
.6.
Hu
, G. D.
, 2014
, “A Modified Version of Explicit Runge–Kutta Methods for Energy-Preserving
,” Kybernetika
, 50
(5
), pp. 838
–847
.https://eudml.org/doc/2621727.
Feng
, K.
, and Shang
, Z.
, 1995
, “Volume-Preserving Algorithms for Source-Free Dynamical Systems
,” Numer. Math.
, 77
(4), pp. 451
–463
.8.
Feng
, K.
, 1992
, “How to Compute Properly Newton's Equation of Motion
,” Second Conference on Numerical Methods for Partial Differential Equations
, L. A.
Ying
and B. Y.
Guo
, eds., World Scientific, Singapore, pp. 15
–22
.9.
Simo
, J. C.
, Tarnow
, N.
, and Wong
, K. K.
, 1992
, “Exact Energy-Momentum Conserving Algorithms and Symplectic Schemes for Nonlinear Dynamics
,” Comput. Methods Appl. Mech. Eng.
, 100
(1
), pp. 63
–116
.10.
Ascher
, U. M.
, and Petzold
, L. R.
, 1998
, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations
, SIAM
, Philadelphia, PA
.11.
McLachlan
, R. I.
, Quispel
, G. R. W.
, and Robidoux
, N.
, 1999
, “Geometric Integration Using Discrete Gradients
,” Philos. Trans. R. Soc. A
, 357
(1754
), pp. 1021
–1046
.12.
Brugnano
, L.
, and Sun
, Y.
, 2014
, “Multiple Invariants Conserving Runge–Kutta Type Methods for Hamiltonian Problems
,” Numer. Algorithms
, 65
(3
), pp. 611
–632
.13.
Terze
, Z.
, Müller
, A.
, and Zlatar
, D.
, 2015
, “An Angular Momentum and Energy Conserving Lie-Group Integration Scheme for Rigid Body Rotational Dynamics Originating From Störmer-Verlet Algorithm
,” ASME J. Comput. Nonlinear Dyn.
, 10
(5
), p. 051005
.14.
Bridges
, T. J.
, 1997
, “Multi-Symplectic Structures and Wave Propagation
,” Math. Proc. Cambridge Philos. Soc.
, 121
(1
), pp. 147
–190
.15.
Hu
, W. P.
, Deng
, Z. C.
, Han
, S. M.
, and Zhang
, W. R.
, 2013
, “Generalized Multi-Symplectic Integrators for a Class of Hamiltonian Nonlinear Wave PDEs
,” J. Comput. Phys.
, 235
, pp. 394
–406
.16.
Hu
, W. P.
, Deng
, Z. C.
, and Yin
, T. T.
, 2017
, “Almost Structure-Preserving Analysis for Weakly Linear Damping Nonlinear Schrödinger Equation With Periodic Perturbation
,” Commun. Nonlinear Sci. Numer. Simul.
, 42
, pp. 298
–312
.17.
Bridges
, T. J.
, and Reich
, S.
, 2001
, “Multi-Symplectic Integrators: Numerical Schemes for Hamiltonian PDEs That Conserve Symplecticity
,” Phys. Lett. A
, 284
(4–5), pp. 184
–193
.18.
Hu
, W. P.
, Deng
, Z. C.
, Wang
, B.
, and Ouyang
, H. J.
, 2013
, “Chaos in an Embedded Single-Walled Carbon Manotube
,” Nonlinear Dyn.
, 72
(1), pp. 389
–398
.19.
Hu
, W. P.
, Deng
, Z. C.
, and Zhang
, Y.
, 2014
, “Multi-Symplectic Method for Peakon-Antipeakon Collision of Quasi-Degasperis-Procesi Equation
,” Comput. Phys. Commun.
, 185
(7
), pp. 2020
–2028
.20.
Hu
, W. P.
, Li
, Q. J.
, Jiang
, X. H.
, and Deng
, Z. C.
, 2017
, “Coupling Dynamic Behaviors of Spatial Flexible Beam With Weak Damping
,” Int. J. Numer. Methods Eng.
, epub.21.
Nocedal
, J.
, and Wright
, S. J.
, 1999
, Numerical Optimization
, Springer-Verlag
, New York
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