One-sided direct event location techniques in the numerical solution of discontinuous differential systems

Abstract

In this paper, event location techniques for a differential system the solution of which is directed towards a manifold $$\varSigma $$ defined as the 0-set of a smooth function $$h: \varSigma =\{x\in \mathbb {R}^n\,:\, h(x)=0 \}$$ are considered. It is assumed that the exact solution trajectory hits $$\varSigma $$ non-tangentially, and numerical techniques guaranteeing that the trajectory approaches $$\varSigma $$ from one side only (i.e., does not cross it) are studied. Methods based on Runge Kutta schemes which arrive to $$\varSigma $$ 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.