One-Sided Direct Event Location Techniques in the Numerical Solution of Discontinuous Differential Systems

In this short paper, event location techniques for a differential system the solution of which is directed towards a surface S defined as the 0-set of a smooth function h: S = {x\in R^n : h(x) = 0 } are considered. It is assumed that the exact solution trajectory hits S non-tangentially, and numerical techniques guaranteeing that the trajectory approaches S from one side only (i.e., does not cross it) are studied. Methods based on Runge Kutta schemes which arrive to S in a finite number of steps are proposed. The main motivation of this paper comes from integration of discontinuous differential systems of Filippov type, where location of events is of paramount importance.