• Journal of information and communication convergence engineering
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• v.12 no.1
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• pp.39-45
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• 2014

A numerical method for solving Hamiltonian equations is said to be symplectic if it preserves the symplectic structure associated with the equations. Various symplectic methods are widely used in many fields of science and technology. A symplectic method preserves an approximate Hamiltonian perturbed from the original Hamiltonian. It theoretically supports the effectiveness of symplectic methods for long-term integration. Although it is also related to long-term integration, numerical stability of symplectic methods have received little attention. In this paper, we consider explicit symplectic methods defined for Hamiltonian equations with Hamiltonians of the special form, and study their numerical stability using the harmonic oscillator as a test equation. We propose a new stability criterion and clarify the stability of some existing methods that are visually based on the criterion. We also derive a new method that is better than the existing methods with respect to a Courant-Friedrichs-Lewy condition for hyperbolic equations; this new method is tested through a numerical experiment with a nonlinear wave equation.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 2006.04a
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• pp.389-394
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• 2006

Fully flexible cell preserves Hamiltonian in structure, so the symplectic time integrator is applied to the equations of motion. Primarily, generalized leapfrog time integration (GLF) is applicable, but the equations of motion by GLF have some of implicit formulas. The implicit formulas give rise to a complicate calculation for coding and need an iteration process. In this paper, the time integration formulas are obtained for the fully flexible cell molecular dynamics simulation by using the splitting time integration. It separates flexible cell Hamiltonian into terms corresponding to each of Hamiltonian term, so the simple and completely explicit recursion formula was obtained. The explicit formulas are easy to implementation for coding and may be reduced the integration time because they are not need iteration process. We are going to compare the resulting splitting time integration with the implicit generalized leapfrog time integration.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.31 no.8
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• pp.826-832
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• 2007

Fully flexible cell preserves Hamiltonian in structure so that the symplectic time integrator is applicable to the equations of motion. In the direct formulation of fully flexible cell N-Sigma-T ensemble, a generalized leapfrog time integration (GLF) is applicable for fully flexible cell simulation, but the equations of motion by GLF has structure of implicit algorithm. In this paper, the time integration formula is derived for the fully flexible cell molecular dynamics simulation by using the splitting time integration. It separates flexible cell Hamiltonian into terms corresponding to each of Hamiltonian term. Thus the simple and completely explicit recursion formula was obtained. We compare the performance and the result of present splitting time integration with those of the implicit generalized leapfrog time integration.

• Bulletin of the Korean Chemical Society
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• v.27 no.12
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• pp.1976-1980
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• 2006

We have calculated the reaction probability and the reaction cross-section of the $N(^4S)+O_2(X^3\sum_{g}^{-})\;\rightarrow\;NO(X^2\Pi)+O(^3P)$ reaction by the quasiclassical trajectory method with the 6th-order explicit symplectic algorithm, based on a new ground potential energy surface. The advantage of the 6th-order explicit symplectic algorithm, conserving both the total energy and the total angular momentum of the reaction system during the numerical integration of canonical equations, has firstly analyzed in this work, which make the calculation of the reaction probability more reliable. The variation of the reaction probability with the impact parameter and the influence of the relative translational energy on the reaction cross-section of the reaction have been discussed in detail. And the fact is found by the comparison that the reaction probability and the reaction cross-section of the reaction estimated in this work are more reasonable than the theoretical ones determined by Gilibert et al.

• Proceedings of the KSME Conference
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• 2005.11a
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• pp.232.1-232.1
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• 2005

• Proceedings of the KSME Conference
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• 2007.05a
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• pp.512-517
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• 2007

In the previous development of the recursive thermostat chained fully flexible cell molecular dynamics simulation, implicit time integration method such as generalized leapfrog integration is used. The implicit algorithm is very much complicated and not easy to show time reversibility because it is solved by the nonlinear iterative procedure. Thus we develop simple, explicit symplectic time integration formula for the recursive thermostat chained fully flexible unit cell simulation. Uniaxial tension test is performed to verify the present explicit algorithm. We check that the present simulation satisfies the ergodic hypothesis for various values of fictitious mass and coefficient of multiple thermostat system. The proposed method should be helpful to predict mechanical and thermal behavior of nano-scale structure.