Symmetric second derivative integration methods

The purpose of this paper is the derivation of symmetric second derivative general linear methods (SGLMs) for general time-reversible differential equations. For this purpose, we find algebraic conditions on the coefficients matrices of the methods which are sufficient for the method to be symmetric. Some symmetric methods of order 4 are constructed and implemented on the famous reversible problems. Numerical experiments show that the constructed symmetric SGLMs approximately conserve the invariants of motion over long time intervals for reversible Hamiltonian systems.