One-sided direct event location techniques in the numerical solution of discontinuous differential systems (Q906947)

In numerical integration of discontinuous systems of Filippov type, as those arising in control engineering, it is essential that the computed trajectory of the ordinary differential equation hits correctly the discontinuity manifold (to avoid, for instance, undesired numerical chattering phenomena). This paper proposes a technique for approaching the event manifold from one side in such a way that only a finite number of steps is required to hit it. The idea is to use a time reparametrization in the original system of differential equations and then use a Runge-Kutta (RK) scheme in the new formulation. Since RK methods preserve linear invariants, then the numerical trajectory automatically reaches exactly a planar manifold in a finite number of steps. This is also true for RK Gauss methods when the manifold is quadratic. In the general case, it is shown that the scheme can be made monotone for a sufficiently small step size. Numerical tests show that RK methods on the transformed problem give generally better results than in the original formulation.