Deterministic particle approximation of aggregation-diffusion equations on unbounded domains

We consider a one-dimensional aggregation-diffusion equation, which is the gradient flow in the Wasserstein space of a functional with competing attractive-repulsive interactions. We prove that the fully deterministic particle approximations with piecewise constant densities introduced in [25] starting from general bounded initial densities converge strongly in to the unique bounded weak solution of the PDE. In particular, the result is achieved in unbounded domains and for arbitrary nonnegative bounded initial densities, thus extending the results in [30], [34], [35] (in which a no-vacuum condition is required) and giving an alternative approach to [10] in the one-dimensional case, including also subquadratic and superquadratic diffusions. We provide also numerical simulations for the approximation scheme.