In this paper we develop one dimensional non-linear ADER schemes for the shallow water system with source terms. In contrast to conventional schemes we adopt a formulation given in terms of the free surface elevation and water discharge. Data reconstruction is performed using ENO polynomials both for the conservative variables and for the bottom elevation. The scheme is accurate up to forth order both in time and in space; essentially non-oscillatory results are obtained for discontinuous solutions both for the steady and unsteady case. The resulting numerical schemes can be applied to several realistic cases characterized by non-uniform geometries.