Structure-preserving properties of three differential schemes for oscillator system

A numerical method for the Hamiltonian system is required to preserve some structure-preserving properties. The current structure-preserving method satisfies the requirements that a symplectic method can preserve the symplectic structure of a finite dimension Hamiltonian system, and a multi-symplectic method can preserve the multi-symplectic structure of an infinite dimension Hamiltonian system. In this paper, the structure-preserving properties of three differential schemes for an oscillator system are investigated in detail. Both the theoretical results and the numerical results show that the results obtained by the standard forward Euler scheme lost all the three geometric properties of the oscillator system, i.e., periodicity, boundedness, and total energy, the symplectic scheme can preserve the first two geometric properties of the oscillator system, and the Störmer-Verlet scheme can preserve the three geometric properties of the oscillator system well. In addition, the relative errors for the Hamiltonian function of the symplectic scheme increase with the increase in the step length, suggesting that the symplectic scheme possesses good structure-preserving properties only if the step length is small enough.