Composite symmetric second derivative general linear methods for Hamiltonian systems

Symmetric second derivative general linear methods (SGLMs) have been already introduced for the numerical solution of time-reversible differential equations. To construct suitable high order methods for such problems, the newly developed composition theory has been successfully used for structure-preserving methods. In this paper, composite symmetric SGLMs are introduced using the generalization of composite theory for general linear methods. Then, we construct symmetric methods of order six by the composition of symmetric SGLMs of order four. Numerical results of the constructed methods verify the theoretical order of accuracy and illustrate that the invariants of motion over long time intervals for reversible Hamiltonian systems are well preserved.