A convergent convex splitting scheme for a nonlocal cahn-hilliard-oono type equation with a transport term

Laurence Cherfils (1) , Hussein Fakih (2, 3) , Maurizio Grasselli (4) , Alain Miranville (5)

1 LaSIE - Laboratoire des Sciences de l'Ingénieur pour l'Environnement - UMR 7356
2 LIU - Lebanese International University
3 LU / ULB - الجامعة اللبنانية [بيروت] = Lebanese University [Beirut] = Université libanaise [Beyrouth]
4 POLIMI - Politecnico di Milano [Milan]
5 LMA [Poitiers] - Laboratoire de mathématiques et applications [UMR 7348]

We devise a rst-order in time convex splitting scheme for a nonlocal Cahn HilliardOono type equation with a transport term and subject to homogeneous Neu-mann boundary conditions. The presence of the transport term is not a minor modi-cation, since, for instance, we lose the unconditional unique solvability and stability. However, we prove the stability of our scheme when the time step is suciently small. Furthermore, we prove the consistency of this scheme and the convergence to the exact solution. Finally, we give some numerical simulations which conrm 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.

Domaines: Mathématiques [math]; Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]; Physique [physics]; Physique mathématique [math-ph]

Laurence Cherfils, Hussein Fakih, Maurizio Grasselli, Alain Miranville. A convergent convex splitting scheme for a nonlocal cahn-hilliard-oono type equation with a transport term. 2018. ⟨hal-01826652⟩