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Research Papers

A Modified Runge–Kutta Method for Nonlinear Dynamical Systems With Conserved Quantities

[+] Author and Article Information
Guang-Da Hu

Professor
Department of Mathematics,
Shanghai University,
Shanghai 200444, China
e-mail: ghu@hit.edu.cn

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received November 28, 2016; final manuscript received May 10, 2017; published online July 12, 2017. Assoc. Editor: Haiyan Hu.

J. Comput. Nonlinear Dynam 12(5), 051026 (Jul 12, 2017) (7 pages) Paper No: CND-16-1586; doi: 10.1115/1.4036761 History: Received November 28, 2016; Revised May 10, 2017

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.

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References

Figures

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Fig. 1

The projection of the phase trajectory on the ω1ω2 plane for h = 0.05

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Fig. 2

The projection of the phase trajectory on the ω1ω3 plane for h = 0.05

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Fig. 3

The projection of the phase trajectory on the ω2ω3 plane for h = 0.05

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Fig. 4

The energy function E1(t) for h = 0.05

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Fig. 5

The angular momentum function E2(t) for h = 0.05

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Fig. 6

αn||∇E(xn+1)|| for h = 0.05

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Fig. 7

The projection of the phase trajectory on the ω1ω2 plane for h = 0.1

Grahic Jump Location
Fig. 8

The projection of the phase trajectory on the ω1ω3 plane for h = 0.1

Grahic Jump Location
Fig. 9

The projection of the phase trajectory on the ω2ω3 plane for h = 0.1

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Fig. 10

The energy function E1(t) for h = 0.1

Grahic Jump Location
Fig. 11

The angular momentum function E2(t) for h = 0.1

Grahic Jump Location
Fig. 12

αn||∇E(xn+1)|| for h = 0.1

Grahic Jump Location
Fig. 13

The projection of the phase trajectory on the ω1ω2 plane for h = 0.2

Grahic Jump Location
Fig. 14

The projection of the phase trajectory on the ω1ω3 plane for h = 0.2

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Fig. 15

The projection of the phase trajectory on the ω2ω3 plane for h = 0.2

Grahic Jump Location
Fig. 16

The energy function E1(t) for h = 0.2

Grahic Jump Location
Fig. 17

The angular momentum function E2(t) for h = 0.2

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Fig. 18

αn||∇E(xn+1)|| for h = 0.2

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