Fully-discrete decoupled Subdivision-based IGA-IEQ-ZEC numerical scheme for the binary surfactant phase-field model coupled with Darcy flow equations on Surfaces

In this paper, we present a comprehensive numerical investigation of the binary phase-field surfactant model coupled with the Darcy flow equation to explore the impact of surfactant addition on the evolution of Saffman–Taylor fingering patterns within a Hele-Shaw cell on surfaces. We develop an efficient and robust spatiotemporal discretization framework that effectively addresses the highly nonlinear terms arising from the strong coupling structure inherent to the model on surface geometries. For the spatial discretization, we employ the recently developed subdivision-based isogeometric analysis (IGA), which provides the advantages of hierarchical refinability and adaptability to arbitrary topologies. This approach eliminates geometric errors associated with surface approximation and reduces additional approximation errors introduced by the numerical schemes. For the temporal discretization, we integrate the Invariant Energy Quadratization (IEQ) method – used to linearize the nonlinear potential – with the Zero-Energy-Contribution (ZEC) decoupling approach, which facilitates fully decoupled computations. The resulting fully discrete numerical framework possesses several desirable properties, including geometric exactness, compatibility with arbitrary topologies, linearity, second-order temporal accuracy, full decoupling, and unconditional energy stability. Additionally, we rigorously establish the unconditional energy stability of the scheme within this work. Furthermore, we perform several numerical experiments to demonstrate the accuracy and robustness of our method, including simulations of benchmark Saffman–Taylor fingering instability to evaluate the weakening effects of surfactants on surface tension.