Error Estimates and Adaptive Time-Step Control for a Class of One-Step Methods for Stiff Ordinary Differential Equations

We prove new optimal a priori error estimates for a class of implicit one-step methods for stiff ordinary differential equations obtained by using the discontinuous Galerkin method with piecewise polynomials of degree zero and one. Starting from these estimates we propose a new algorithm for automatic time-step control and we discuss the relation between this algorithm and earlier algorithms implemented in packages for the numerical solution of stiff ordinary differential equations.