Consistency-enhanced E-SAV BDF2 time-marching method with relaxation for the hydrodynamically-coupled binary phase-field crystal model

This paper focuses on the hydrodynamically-coupled binary phase-field crystal (BPFC) model. Based on the $L^2$-gradient flow, we derive the governing equations from the energy functional. To ensure the conservation of total mass, we incorporate two nonlocal Lagrange multipliers. For fluid dynamics, we employ the incompressible Navier–Stokes (NS) equations. Subsequently, we analytically prove the crucial property of energy dissipation within the model. To devise the numerical scheme, we reformulate the governing equations into an equivalent representation by incorporating an exponential scalar auxiliary variable (E-SAV). By leveraging the second-order backward difference formula (BDF2), we initially devise a second-order scheme and subsequently correct the energy through a relaxation technique applied to the E-SAV, with the objective of enhancing energy consistency. At every time step, due to the complete decoupling of variables, we solve only a limited number of elliptic equations to update the relevant variables. Furthermore, we analyze the unique solvability and energy stability of the time-discretized method. To confirm the performance of our proposed scheme, we perform various numerical experiments aimed at validating its effectiveness, accuracy, and stability.