Double Fourier Sphere Methods With Low Rank Approximation for Block Copolymer Systems on Sphere

ABSTRACT We introduce Double Fourier Sphere (DFS) methods for the Ohta–Kawasaki (OK) and Nakazawa–Ohta (NO) models on a spherical domain, examining their coarsening dynamics and equilibrium pattern formations. We employed DFS for spatial discretization and the second‐order Backward Differentiation Formula (BDF2) scheme for time evolution, resulting in an efficient energy‐stable scheme to simulate the OK and NO models on the unit sphere. Our numerical experiments reveal various self‐assembled patterns, such as single‐bubble assemblies in binary systems and double‐bubble and mixed‐bubble assemblies in ternary systems. These patterns closely resemble experimental biomembrane patterns, demonstrating the effectiveness of the OK model in real‐world applications. Additionally, our study explores the relationship between repulsive strength and the number of bubbles in assemblies, confirming the two‐thirds law in the OK model. This provides quantitative evidence of how self‐assembled patterns depend on system parameters in copolymer systems.