Efficient solution of IVPs with right-hand sides having discontinuities on an unknown hypersurface

摘要

The paper deals with the solution of systems of IVPs with discontinuous right-hand side functions f. In our previous work we considered IVPs with continuous f ’s having discontinuous derivatives. We assumed the global Lipschitz continuity of f, and the analysis was heavily based on that assumption. In the present paper we skip the continuity assumption. We assume that f is a piecewise regular function that satisfies the Lipschitz condition only in some regions of the domain. (By the regular function we mean a function belonging to a Hölder class.) A hypersurface where the regularity of f breaks down is not known. We define an algorithm DISC-IVP for solving such problems, and analyze its error and cost. The error analysis in the presence of discontinuities of f is essentially different than that in the continuous case. We show in a subclass of f ’s that the worst-case error of the algorithm (measured in the conservative supremum norm over the interval of integration) is of the same order as for globally regular functions. The algorithm and the analysis do not use heuristic arguments. In particular, we are able to bound the cost of the algorithm, which leads to the bound on the ε-complexity of the problem (the minimal cost of computing an ε-approximation to the solution). The ε-complexity in the discontinuous case described above turns out to be of the same order as that in the globally regular case. In the other words, the solution of discontinuous problems is no more difficult than the solution of globally regular ones. Furthermore, the algorithm DISC-IVP is error and cost optimal.