Collective Behavior: Macroscopic versus Kinetic Descriptions

A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure

Yanghong Huang

Imperial College London

We introduce a positivity preserving entropy decreasing finite volume scheme for nonlinear nonlocal equations with a gradient flow structure. These properties allow for accurate computations of stationary states and long-time asymptotics of various systems, including macroscopic equations describing collective behaviours. 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.