Ultraspherical Spectral Method for Block Copolymer Systems on Unit Disk (Advances in Applied Mathematics and Mechanics, 2024)
In this paper, we investigate ultraspherical spectral method for the Ohta-Kawasaki (OK) and Nakazawa-Ohta (NO) models in the disk domain, representingdiblock and triblock copolymer systems, respectively. We employ ultraspherical spectral discretization for spatial variables in the disk domain and apply the second-orderbackward differentiation formula (BDF) method for temporal discretization. To ourbest knowledge, this is the first study to develop a numerical method for diblock andtriblock copolymer systems with long-range interactions in disk domains. We showthe energy stability of the numerical method in both semi-discrete and fully-discretediscretizations. In our numerical experiments, we verify the second-order temporalconvergence rate and the energy stability of the proposed methods. Our numericalresults show that the coarsening dynamics in diblock copolymers lead to bubble assemblies both inside and on the boundary of the disk. Additionally, in the triblockcopolymer system, we observe several novel pattern formations, including single anddouble bubble assemblies in the unit disk. These findings are detailed through extensive numerical experiments.