Fast, unconditionally energy stable large time stepping method for a new Allen–Cahn type square phase-field crystal model

Weak solutions and simulations to a square phase‐field crystal model with Neumann boundary conditions

“…The snapshots of the profiles of the order parameter ϕ at various times are shown in Figure 3. From the numerical results, we observe that the model finally generates the square symmetry patterns in the domain, which is qualitatively consistent with the simulations shown in Lin et al 25,27 From the previous discussion, we know that there are many approaches to obtain a square lattice, but these methods require higher order derivatives or even nonlinear higher order terms in the free energy, which bring great difficulties to numerical simulation. Compared with other extensions of the PFC model, the SPFC model provides a simpler method to obtain square lattices.…”

“…The only difference between the PFC and SPFC equations is the replacement of ϕ 4 in () with |∇ ϕ | 4 , while mathematical analysis of the later one is more challenging. This equation () has been well studied in other studies 23–27 from the numerical analysis point of view. For example, Cheng et al 24 considered two energy stable schemes for the SPFC equation, both with second order temporal accuracy and Fourier pseudo‐spectral spatial discretization and designed numerical algorithms based on the structures of the individual energy terms to overcome the difficulties associated with this highly nonlinear operator.…”

“…They developed a general method for constructing free energy functionals that exhibit solid-liquid coexistence with desired crystal symmetries. Inspired by the work of other studies, 3,22,23 Cheng et al [24][25][26] introduced the SPFC equation, which is a simpler method to obtain a square symmetry crystal lattice.…”

Efficient unconditionally stable numerical schemes for a modified phase field crystal model with a strong nonlinear vacancy potential

“…In Reference [5], the authors proposed an energy stable second-order backward differentiation formulas (BDF2) Fourier pseudo-spectral numerical scheme for the SPFC model [4,15], using the preconditioned steepest descent (PSD) iteration [14] to solve the corresponding nonlinear system. The authors in Reference [35] developed a stabilized SAV scheme for a new Allen-Cahn type SPFC model. For the MPFC model, many numerical strategies have also been applied to construct energy stable numerical schemes, for instance, the convex splitting methods [1,2,29,52], the IEQ approach [30], the SAV approach [31], and many others [9,23,25].…”

Discontinuous finite volume element method for a coupled Navier-Stokes-Cahn-Hilliard phase field model

“…Moreover, the presence of the energy law serves as a guide line for the design of energy stable numerical schemes. Various numerical methods have been developed and analyzed for different phase field models, such as the finite element method [3,30,35,40,45,52,54,100], finite difference method [14,16,99], spectral method [46,67,87,91,102], extended finite element method [18,32], discontinuous Galerkin finite element method [31,66,84], finite volume method [7,106], penalty-projection method [83], lattice Boltzmann method [26,107], and many others [13,50,53,63,71,77,79,80,98,105].…”