Nonlinear nonlocal equations are more and more popular the modeling of many physical and biological processes, like self-assembly of nanoparticles and collective behaviors of animals. In this talk, we focus on a finite volume scheme for equations that have a gradient flow structure, along which the entropy is decreasing. Built upon ideas from conservation laws, the scheme preserves the non-negativity of the solutions and the structure of the gradient flow. These properties allow for accurate computations of stationary states and long-time asymptotics of various systems. The proposed scheme is able to cope with non-smooth stationary states, different time scales including metastability, as well as concentrations and self-similar behavior induced by singular nonlocal kernels. We use the scheme to explore properties of these equations beyond their present theoretical knowledge. This is a joint work with Jose Carrillo and Alina Chertock.