A general symplectic integrator for canonical Hamiltonian systems
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The focus of this paper is to recommend a novel symplectic scheme for canonical Hamiltonian systems. The new scheme contains a real parameter which makes the symplectic Euler methods and implicit midpoint rule as its special cases. The validity of the symplecticity of this scheme is well explained from the perspectives of the generating function and partitioned Runge-Kutta methods. The generating function of the new symplectic scheme with new coordinates is studied, and these coordinates include the three typical coordinates. Employing the generating function method and symmetric composition methods, two new classes of symplectic schemes of any high order are devised, respectively. Furthermore, based on these symplectic schemes, two energy-preserving schemes and the feasibility of constructing higher order energy-preserving schemes are presented by the parameter serving for clever tuning. The solvability of all the schemes mentioned is proved, and the numerical performances of these schemes are demonstrated with numerical experiments.
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