The cohesiveness of G-symplectic methods

Abstract

General linear methods are multistage multivalue methods. This large family of numerical methods for ordinary differential equations, includes Runge---Kutta and linear multistep methods as special cases. G-symplectic general linear methods are multivalue methods which preserve a generalization of quadratic invariants. If Q is an invariant quadratic form then symplectic Runge---Kutta methods preserve this invariant. In the case of a G-symplectic general linear method, there exists a non-singular symmetric r×r matrix G such that G?Q is an invariant quadratic form for this method. Although the numerical results can be corrupted by parasitic behaviour, it is possible to overcome the effect of parasitic growth by imposing additional constraints on the method. For G-symmetric methods satisfying these additional conditions, numerical experiments give excellent results. A new concept known as "cohesiveness" is introduced in an attempt to explain this favourable numerical behaviour. It is shown that the deviation from perfect cohesiveness grows slowly as steps of the method are carried out.