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

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