Adaptive mesh selection asymptotically guarantees a prescribed local error for systems of initial value problems

We study adaptive mesh selection for the solution of systems of initial value problems. The goal is a rigorous theoretical analysis of potential advantages of adaption. For an optimal method in the sense of the speed of convergence, we propose an algorithm for successive selection of the mesh points. The selection is based on an upper bound on the local error, and it (asymptotically) guarantees the local errors not exceeding a prescribed level. The mesh selection algorithm can be applied to a general class of methods, not only to the chosen one. We rigorously discuss the cost of the proposed algorithm, comparing it to other algorithms equipped with different mesh selection procedures. We specify a quantitative advantage of the adaptive mesh over the uniform one. Adjustment of the mesh points to a local behavior of the solution yields improved efficiency of the algorithm. Some numerical results illustrating theoretical findings are reported.