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.
Keywords: Cahn–Hilliard–Oono equation, transport term, nonlocal term, convex splitting scheme, stability and convergence, numerical simulations
@article{M2AN_2021__55_S1_S225_0,
author = {Cherfils, Laurence and Fakih, Hussein and Grasselli, Maurizio and Miranville, Alain},
title = {A convergent convex splitting scheme for a nonlocal {Cahn{\textendash}Hilliard{\textendash}Oono} type equation with a transport term},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {S225--S250},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {Suppl\'ement},
doi = {10.1051/m2an/2020028},
mrnumber = {4221297},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2020028/}
}
TY - JOUR AU - Cherfils, Laurence AU - Fakih, Hussein AU - Grasselli, Maurizio AU - Miranville, Alain TI - A convergent convex splitting scheme for a nonlocal Cahn–Hilliard–Oono type equation with a transport term JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - S225 EP - S250 VL - 55 IS - Supplément PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2020028/ DO - 10.1051/m2an/2020028 LA - en ID - M2AN_2021__55_S1_S225_0 ER -
%0 Journal Article %A Cherfils, Laurence %A Fakih, Hussein %A Grasselli, Maurizio %A Miranville, Alain %T A convergent convex splitting scheme for a nonlocal Cahn–Hilliard–Oono type equation with a transport term %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P S225-S250 %V 55 %N Supplément %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2020028/ %R 10.1051/m2an/2020028 %G en %F M2AN_2021__55_S1_S225_0
Cherfils, Laurence; Fakih, Hussein; Grasselli, Maurizio; Miranville, Alain. A convergent convex splitting scheme for a nonlocal Cahn–Hilliard–Oono type equation with a transport term. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021), pp. S225-S250. doi: 10.1051/m2an/2020028
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