G-symplectic second derivative general linear methods for Hamiltonian problems

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 44. 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.