Provably third-order energy stable adaptive algorithm for modeling square pattern in phase field crystal

Square pattern emerges widely in crystallography, ranging from soft matters to thermal convection in fluid dynamics. The square phase field crystal equation models such pattern formation on atomic length and diffusive time scales. The governing equation, derived from a conserved gradient flow of a free energy, involves sixth-order spatial derivatives and Laplacian-gradient type nonlinear term, which result in severe stability restriction on the time stepsizes and difficulty in theoretical analysis. In this paper, we propose a novel unconditionally energy stable, third-order adaptive BDF scheme. The convex-splitting and a multistep stabilization are leveraged to maintain energy stable with arbitrary time stepsizes, and the adaptive time-stepping control based on evolution rate is applied to efficiently obtain high-resolution results. We strictly prove optimal error estimate in the variable-step setting under a mild step ratio constraint by enhancing the discrete kernel framework proposed recently. Numerical tests in 2D/3D demonstrate the square pattern evolution in crystallization process from random or supercooled liquid initial states over a long time. The adaptive algorithm reduces computation time by 95 % while capturing details of phases, confirming the effectiveness of our method. This is the first systematic work on adaptive high-order structure-preserving method for square phase field crystal.