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

¹ Université de La Rochelle, Laboratoire des Sciences de l’Ingénieur pour l’Environnement, UMR CNRS 7356, Avenue Michel Crépeau, La Rochelle Cedex F-17042, France
² Lebanese International University, Department of Mathematics and Physics, Beqaa Campus, Lebanon
³ Lebanese University, Department of Mathematics, Beirut, Lebanon
⁴ Politecnico di Milano – Dipartimento di Matematica, Milano I-20133, Italy
⁵ Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348 – SP2MI, Boulevard Marie et Pierre Curie – Téléport 2, Chasseneuil Futuroscope Cedex F-86962, France
⁶ Xiamen University, School of Mathematical Sciences, Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing, Xiamen, Fujian, P.R. China

^* Corresponding author: hussein.fakih@liu.edu.lb

Received: 29 June 2018
Accepted: 17 April 2020

Abstract

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.