Automatic Control and Adaptive Time-Stepping

Adaptive time-stepping is central to the efficient solution of initial value problems in ODEs and DAEs. The error committed in the discretization method primarily depends on the time-step size h, which is varied along the solution in order to minimize the computational effort subject to a prescribed accuracy requirement. This paper reviews the recent advances in developing local adaptivity algorithms based on well established techniques from linear feedback control theory, which is introduced in a numerical context. Replacing earlier heuristics, this systematic approach results in a more consistent and robust performance. The dynamic behaviour of the discretization method together with the controller is analyzed. We also review some basic techniques for the coordination of nonlinear equation solvers with the primary stepsize controller in implicit time-stepping methods.