One-step methods and implicit extrapolation technique for index 1 differential-algebraic systems

In this paper we first formulate and prove a number of theorems concerning the convergence of combined numerical one-step methods for index I differential-algebraic systems. Then, we use these results to justify an implicit extrapolation technique and show their practical importance. Second, we give a theory of adjoint and symmetric one-step methods for differential-algebraic equations and we also determine symmetric methods among Runge-Kutta formulae. We prove that algebraically stable symmetric Runge-Kutta formulae are symplectic and they have a structure which is in some sense similar to the structure of Gauss methods. Finally, we come to the concept of quadratic extrapolation for index I differential-algebraic systems and develop an advanced version of the local-global step size control based on the extrapolation technique. Numerical tests support the theoretical results of the paper.