Collocation method for differential variational inequality problems

In this paper, a Jacobi collocation method is presented for solving differential variational inequalities (DVIs). Differential variational inequalities consist of a differential equation and a variational inequality. A type of Jacobi-Gauss collocation scheme with N knots is applied to the differential part of the problem whereas another type of Jacobi-Gauss collocation scheme with N+1 knots is applied to the variational part of it. So the DVI problem turns into a variational inequality problem. Electrical circuits with nonsmooth elements like ideal diodes are an important class of physical systems, which can be modeled as DVI problems. So in the numerical experiments, 1 example with smooth solutions and 4 illustrative examples of simple electrical circuits with ideal diodes are considered. Numerical results demonstrate the effectiveness of the proposed method but slow convergence for the proposed method for some examples. The reason for slow convergence in this method is that the solutions of these DVIs are nonsmooth.