Ordinary Differential Equations (ODE) with discontinuous right--hand side appear as dynamical models in various fields ranging from automatic control (sliding mode and optimal control, hybrid systems), electrical and mechanical engineering (ideal electrical components, saturation, friction, piecewise linear models) to biology (gene regulatory networks). To deal with ODEs with discontinuities, a concept of solution has to be firstly defined, since standard Caratheodory solutions do no longer apply. Filippov's framework (notion of solutions and extension as differential inclusions) is generally invoked. If the latter is a powerful tool for the existence analysis of solutions, it is not an amenable approach for the numerical computation of solutions. We propose in this work to embed ODEs with discontinuities in the framework of complementarity systems and Differential Variational Inequalities (DVI). We show that this embedding is equivalent to a special differential inclusion extension alternative to the Filippov classical one. This reformulation enables the design of a time--stepping scheme for which is it possible to show that the variational inequality that we have to solve at each time--step is solvable. It also make possible an efficient computation of equilibrium points and theirs stability analysis. The approach will be illustrated on the simulation of gene regulatory networks in cell Biology.