Optimal adaptive solution of piecewise regular systems of IVPs with unknown switching hypersurface

We study the solution of piecewise regular systems of IVPs with a switching hypersurface defined by an unknown event function. Our aim is to design an algorithm that is not based on heuristic steps, it does not use information on the event function and preserves the order known for regular systems. The algorithm is based on the adaptive mesh which differs from the equidistant mesh only by a single point. We provide a rigorous (that is, independent of heuristic arguments) analysis of its error and cost.