Projected Runge-Kutta methods for constrained Hamiltonian systems

doi:10.1007/s10483-016-2119-8

Applied Mathematics and Mechanics (English Edition) ›› 2016, Vol. 37 ›› Issue (8): 1077-1094.doi: https://doi.org/10.1007/s10483-016-2119-8

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Projected Runge-Kutta methods for constrained Hamiltonian systems

Yi WEI¹, Zichen DENG², Qingjun LI², Bo WANG³

1. Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710072, China;
2. Department of Engineering Mechanics, Northwestern Polytechnical University, Xi'an 710072, China;
3. Mechanical and Aerospace Engineering, Oklahoma State University, Stillwater 74074, U. S. A.

Received:2015-07-26 Revised:2016-04-04 Online:2016-08-01 Published:2016-08-01
Contact: Zichen DENG E-mail:dweifan@nwpu.edu.cn
Supported by:
Project supported by the National Natural Science Foundation of China (No. 11432010), the Doctoral Program Foundation of Education Ministry of China (No. 20126102110023), the 111 Project of China (No. B07050), the Fundamental Research Funds for the Central Universities (No. 310201401JCQ01001), and the Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (No.CX201517)

Abstract

Abstract:

Projected Runge-Kutta (R-K) methods for constrained Hamiltonian systems are proposed. Dynamic equations of the systems, which are index-3 differential-algebraic equations (DAEs) in the Heisenberg form, are established under the framework of Lagrangian multipliers. R-K methods combined with the technique of projections are then used to solve the DAEs. The basic idea of projections is to eliminate the constraint violations at the position, velocity, and acceleration levels, and to preserve the total energy of constrained Hamiltonian systems by correcting variables of the position, velocity, acceleration, and energy. Numerical results confirm the validity and show the high precision of the proposed method in preserving three levels of constraints and total energy compared with results reported in the literature.

Key words: differential-algebraic equation (DAE), constraint violation, energy and constraint preservation, constrained Hamiltonian system, projected Runge-Kutta (R-K) method

2010 MSC Number:

74H40
70G45

Yi WEI, Zichen DENG, Qingjun LI, Bo WANG. Projected Runge-Kutta methods for constrained Hamiltonian systems. Applied Mathematics and Mechanics (English Edition), 2016, 37(8): 1077-1094.

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Projected Runge-Kutta methods for constrained Hamiltonian systems

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