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**A precise Runge-Kutta integration and its application for solving nonlinear dynamical systems.**
*(English)*
Zbl 1114.65084

Summary: Precise integration is compounded with the Runge-Kutta method and a new effective integration method is presented for solving nonlinear dynamical system. An arbitrary dynamical system can be expressed as a nonhomogeneous linear differential equation problem. Precise integration is used to solve its corresponding homogeneous equation and Runge-Kutta methods can be used to calculate the nonhomogeneous, nonlinear items. The precise integration may have large time step and the time step of the Runge-Kutta methods can be adjusted to improve the computational precision. The handling technique in this paper not only avoids the matrix inversion but also improves the stability of the numerical method. Finally, the numerical examples are given to demonstrate the validity and effectiveness of the proposed method.

### MSC:

65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

34A34 | Nonlinear ordinary differential equations and systems |

65L20 | Stability and convergence of numerical methods for ordinary differential equations |

### Keywords:

precise integration; nonlinear dynamical equation; Runge-Kutta method; stability; numerical examples
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\textit{S. Zhang} et al., Appl. Math. Comput. 184, No. 2, 496--502 (2007; Zbl 1114.65084)

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### References:

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