A convergent convex splitting scheme for a nonlocal Cahn-Hilliard-Oono type equation with a transport term

We devise a first-order in time convex splitting scheme for a nonlocal Cahn-Hilliard-Oono type equation with a transport term and subject to homogeneous Neumann boundary conditions. However, we prove the stability of our scheme when the time step is sufficiently small, according to the velocity field and the interaction kernel. Furthermore, we prove the consistency of this scheme and the convergence to the exact solution. Finally, we give some numerical simulations which confirm our theoretical results and demonstrate the performance of our scheme not only for phase separation, but also for crystal nucleation, for several choices of the interaction kernel.