A new class of decoupled and energy-stable numerical schemes for the binary phase field crystal model

Abstract

In this paper, we introduce a new class of energy-stable, fully decoupled numerical schemes for the binary phase-field crystal (BPFC) model, which is governed by a pair of coupled sixth-order Cahn--Hilliard-type PDEs. The key idea is to split the mobility operator using an operator splitting method. Our method decouples the two phase variables at each time step, significantly reducing computational complexity while preserving unconditional energy stability. We rigorously establish energy dissipation for both first- and second-order accurate schemes in time. Numerical convergence tests confirm the expected temporal accuracy. A series of numerical examples on regular domains, irregular geometries, surfaces, and 3D domains demonstrates the robustness and flexibility of the proposed schemes in capturing complex phase transitions in binary colloidal crystals. The results demonstrate the applicability of the proposed schemes to the BPFC model for simulating multi-component crystalline structures across a range of model parameters and domain geometries.