TY - JOUR
T1 - Ultraspherical Spectral Method for Block Copolymer Systems on Unit Disk
AU - Luo , Wangbo
AU - Zhao , Yanxiang
JO - Advances in Applied Mathematics and Mechanics
VL - 1
SP - 69
EP - 103
PY - 2024
DA - 2024/12
SN - 17
DO - http://doi.org/10.4208/aamm.OA-2024-0161
UR - https://global-sci.org/intro/article_detail/aamm/23593.html
KW - Block copolymer, Ohta-Kawasaki model, Nakazawa-Ohta model, disk domain, ultraspherical spectral method, bubble assembly, ’Head-to-tail’ pattern.
AB - <p style="text-align: justify;">In this paper, we investigate ultraspherical spectral method for the Ohta-Kawasaki (OK) and Nakazawa-Ohta (NO) models in the disk domain, representing
diblock and triblock copolymer systems, respectively. We employ ultraspherical spectral discretization for spatial variables in the disk domain and apply the second-order
backward differentiation formula (BDF) method for temporal discretization. To our
best knowledge, this is the first study to develop a numerical method for diblock and
triblock copolymer systems with long-range interactions in disk domains. We show
the energy stability of the numerical method in both semi-discrete and fully-discrete
discretizations. In our numerical experiments, we verify the second-order temporal
convergence rate and the energy stability of the proposed methods. Our numerical
results show that the coarsening dynamics in diblock copolymers lead to bubble assemblies both inside and on the boundary of the disk. Additionally, in the triblock
copolymer system, we observe several novel pattern formations, including single and
double bubble assemblies in the unit disk. These findings are detailed through extensive numerical experiments.</p>