Optimal adaptive implicit time stepping

We revisit adaptive time stepping, one of the classical topics of numerical analysis and computational engineering. While widely used in application and subject of many theoretical works, a complete understanding is still missing. Apart from special cases, there does not exist a complete theory that shows how to choose the time steps such that convergence towards the exact solution is guaranteed with the optimal convergence rate. In this work, we use recent advances in adaptive mesh refinement to propose an adaptive time stepping algorithm that is mathematically guaranteed to be optimal in the sense that it achieves the best possible convergence of the error with respect to the number of time steps, and it can be implemented using a time stepping scheme as a black box.