Symmetric methods for reversible Hamiltonian problems

Symmetric property of the numerical methods plays a central role to have a good long-time behaviour when they applied to time-reversible ordinary differential equations (ODEs). In this paper, we introduce a class of efficient methods based on second derivative methods for solving time-reversible ODEs. The proposed methods preserve reversibility property of the problems. Furthermore, they have high accuracy comparing with the existing similar methods which it is shown by some numerical evidence