Symmetric general linear methods

Abstract

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.