Structure-Preserving Algorithms for the Lorenz System
GANG Tie-Qiang1, MEI Feng-Xiang1, CHEN Li-Jie2
1Department of Mechanics, Beijing Institute of Technology, Beijing 1000812Department of Mechanical and Electrical Engineering, Xiamen University,Xiamen 361005
Structure-Preserving Algorithms for the Lorenz System
GANG Tie-Qiang1;MEI Feng-Xiang1;CHEN Li-Jie2
1Department of Mechanics, Beijing Institute of Technology, Beijing 1000812Department of Mechanical and Electrical Engineering, Xiamen University,Xiamen 361005
摘要Based on a splitting method and a composition method, we construct some structure-preserving algorithms with first-order, second-order and fourth-order accuracy for a Lorenz system. By using the Liouville's formula, it is proven that the structure-preserving algorithms exactly preserve the volume of infinitesimal cube for the Lorenz system. Numerical experimental results illustrate that for the conservative Lorenz system, the qualitative behaviour of the trajectories described by the classical explicit fourth-order Runge--Kutta (RK4) method and the fifth-order Runge--Kutta--Fehlberg (RKF45) method is wrong, while the qualitative behaviour derived from the structure-preserving algorithms with different orders of accuracy is correct. Moreover, for the small dissipative Lorenz system, the norm errors of the structure-preserving algorithms in phase space are less than those of the Runge--Kutta methods.
Abstract：Based on a splitting method and a composition method, we construct some structure-preserving algorithms with first-order, second-order and fourth-order accuracy for a Lorenz system. By using the Liouville's formula, it is proven that the structure-preserving algorithms exactly preserve the volume of infinitesimal cube for the Lorenz system. Numerical experimental results illustrate that for the conservative Lorenz system, the qualitative behaviour of the trajectories described by the classical explicit fourth-order Runge--Kutta (RK4) method and the fifth-order Runge--Kutta--Fehlberg (RKF45) method is wrong, while the qualitative behaviour derived from the structure-preserving algorithms with different orders of accuracy is correct. Moreover, for the small dissipative Lorenz system, the norm errors of the structure-preserving algorithms in phase space are less than those of the Runge--Kutta methods.
(Ordinary and partial differential equations; boundary value problems)
引用本文:
GANG Tie-Qiang;MEI Feng-Xiang;CHEN Li-Jie. Structure-Preserving Algorithms for the Lorenz System[J]. 中国物理快报, 2008, 25(3): 866-869.
GANG Tie-Qiang, MEI Feng-Xiang, CHEN Li-Jie. Structure-Preserving Algorithms for the Lorenz System. Chin. Phys. Lett., 2008, 25(3): 866-869.
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2008, Vol. 25 
