In this paper we develop a theory of adjoint and symmetric methods in the class of general implicit one-step fixed-stepsize methods for solving differential-algebraic equations. We prove a number of theorems for methods having these properties and show that many results of the theory of adjoint and symmetric one-step fixed-stepsize methods for ordinary differential equations hold in the case of methods for differential-algebraic equations. In particular, we prove that only the symmetric methods possess a quadratic asymptotic expansion of the global error and provide a basis for \tau^2-extrapolation.

CEMAT - Center for Computational and Stochastic Mathematics