G-symplectic second derivative general linear methods for Hamiltonian problems

摘要

It is our purpose to design second derivative general linear methods (SGLMs) for solving Hamiltonian problems. To do this, we explore G-symplectic SGLMs which preserve a generalization of quadratic invariants along the long-time integration. We find sufficient conditions on the coefficients matrices of the methods which ensure G-symplecticity and control parasitism. We construct such methods up to order 4. Numerical experiments of the constructed methods on the well-known Hamiltonian problems indicate ability of the methods in solving Hamiltonian problems over long-time integration.