In this paper, we developed a theoretical study for nonconservative sytems in one dimension in order to construct numerical schemes for solving the Riemann problem. The nonconservative form of our model system required the use of a well-adapted theory in order to give us a sense of our problem. We chose a framework of generalized functions for solving a scalar hyperbolic equation with a discontinuous coefficient $\sigma_t +u\sigma_x \approx 0$, where u is the velocity solution of a Burgers's equation. After an explicit solution of the Riemann problem, we derived Godunov split schemes for computing an approximate solution of the Cauchy problem. We applied our study to a system modeling elasticity and a system modeling gas dynamics. Some stability properties of a scheme and its convergence to a generalized solution are proved for the first model. Numerical experiments confirmed this convergence result. For the second model, calculations of flows containing weak-to-moderate shocks showed that conservation errors are reduced when the mesh is refined but were not entirely eliminated.