Unconditionally energy stable high-order convex-splitting schemes for the square phase field crystal model with complex nonlinearity

This study presents the high-order schemes with unconditional energy stability for solving the square phase field crystal model. The proposed schemes couple BDF$q$ ($q$=3,4,5) method with the convex-splitting strategy, incorporating the multi-time-level stabilization term, which is closely reformed by BDF$q$ method. This newly introduced stabilization term acts as the additional diffusivity, while ensuring the unconditional energy dissipation and the optimal error analysis. Theoretical analysis confirms that the high-order convex-splitting schemes also preserve unique solvability and mass conservation. Notably, the unconditional energy stability guarantees the boundedness of the numerical solution in the discrete $H_h^2$ and $W_h^{1,6}$ norms, which allow precise estimation of nonlinear term via Young’s inequality, overcoming the analytical challenges brought by high-order nonlinear term. Consequently, the optimal error estimate is rigorously conducted using the global energy analysis technology based on the discrete orthogonal convolution kernels. Numerical examples are performed to validate the efficiency and accuracy of the proposed schemes, demonstrating that 5th-order scheme achieves enhanced accuracy especially with large time-step size. The distinct advantages of the proposed high-order schemes are particularly beneficial in long-term simulations. In addition, numerical tests confirm that the relatively large stabilization term $S=5$ is essential for ensuring energy stability and reducing computational costs by $75\%$ compared to $S=1$.