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.

翻译：本文研究了圆盘域中分别代表二嵌段和三嵌段共聚物系统的Ohta-Kawasaki (OK) 和 Nakazawa-Ohta (NO) 模型的超球谱方法。我们对圆盘域中的空间变量采用超球谱离散化，并对时间离散化应用二阶后向差分公式 (BDF) 方法。据我们所知，这是首次针对圆盘域中具有长程相互作用的二嵌段和三嵌段共聚物系统开发数值方法的研究。我们证明了该数值方法在半离散和全离散格式下的能量稳定性。在我们的数值实验中，我们验证了所提方法的二阶时间收敛率和能量稳定性。我们的数值结果表明，二嵌段共聚物中的粗化动力学导致圆盘内部和边界上均形成气泡组装体。此外，在三嵌段共聚物系统中，我们观察到了几种新颖的图案形成，包括单位圆盘内的单气泡和双气泡组装体。这些发现通过大量的数值实验得到了详细阐述。