Revision of the theory of symmetric one-step methods for ordinary differential equations

In this paper we develop a new theory of adjoint and symmetric methods in the class of general implicit one-step fixedstepsize methods. These methods arise from simple and natural definitions of the concepts of symmetry and adjointness that provide a fruitful basis for analysis. We prove a number of theorems for methods having these properties and show, in particular, that only the symmetric methods possess a quadratic asymptotic expansion of the global error. In addition, we give a very simple test to identify the symmetric methods in practice.