Symmetric general linear methods

The article considers symmetric general linear methods, a class of numerical time integration methods which, like symmetric Runge–Kutta methods, are applicable to general time–reversible differential equations, not just those derived from separable second–order problems. A definition of time–reversal symmetry is formulated for general linear methods, and criteria are found for the methods to be free of linear parasitism. It is shown that symmetric parasitism–free methods cannot be explicit, but a method of order $4$ is constructed with only one implicit stage. Several characterizations of symmetry are given, and connections are made with $G$–symplecticity. Symmetric methods are shown to be of even order, a suitable symmetric starting method is constructed and shown to be essentially unique. The underlying one–step method is shown to be time–symmetric. Several symmetric methods of order $4$ are constructed and implemented on test problems. The methods are efficient when compared with Runge–Kutta methods of the same order, and invariants of the motion are well–approximated over long time intervals.