An Energy Stable BDF2 Fourier Pseudo-Spectral Numerical Scheme for the Square Phase Field Crystal Equation

Abstract

In this paper we propose and analyze an energy stable numerical scheme for the square phase field crystal (SPFC) equation, a gradient flow modeling crystal dynamics at the atomic scale in space but on diffusive scales in time. In particular, a modification of the free energy potential to the standard phase field crystal model leads to a composition of the 4-Laplacian and the regular Laplacian operators. To overcome the difficulties associated with this highly nonlinear operator, we design numerical algorithms based on the structures of the individual energy terms. A Fourier pseudo-spectral approximation is taken in space, in such a way that the energy structure is respected, and summation-by-parts formulae enable us to study the discrete energy stability for such a high-order spatial discretization. In the temporal approximation, a second order BDF stencil is applied, combined with an appropriate extrapolation for the concave diffusion term(s). A second order artificial Douglas-Dupont-type regularization term is added to ensure energy stability, and a careful analysis leads to the artificial linear diffusion coming at an order lower that that of surface diffusion term. Such a choice leads to reduced numerical dissipation. At a theoretical level, the unique solvability, energy stability are established, and an optimal rate convergence analysis is derived in the ${\ell }^{\infty }(0,T;{\ell }^2)\cap {\ell }^2(0,T;H_N^3)$ norm. In the numerical implementation, the preconditioned steepest descent (PSD) iteration is applied to solve for the composition of the highly nonlinear 4-Laplacian term and the standard Laplacian term, and a geometric convergence is assured for such an iteration. Finally, a few numerical experiments are presented, which confirm the robustness and accuracy of the proposed scheme.

References in corpus (9)

• Modeling Elasticity in Crystal Growth
• Modeling Elastic and Plastic Deformations in Non-Equilibrium Processing Using Phase Field Crystals
• Phase-field Crystals with Elastic Interactions
• A second order in time, uniquely solvable, unconditionally stable numerical scheme for Cahn-Hilliard-Navier-Stokes equation
• Energy Stable and Efficient Finite-Difference Nonlinear Multigrid Schemes for the Modified Phase Field Crystal Equation
• Convergence Analysis for Second Order Accurate Convex Splitting Schemes for the Periodic Nonlocal Allen-Cahn and Cahn-Hilliard Equations
• Preconditioned Steepest Descent Methods for some Nonlinear Elliptic Equations Involving p-Laplacian Terms
• Controlling crystal symmetries in phase-field crystal models
• A Fourier Pseudospectral Method for the "Good" Boussinesq Equation with Second Order Temporal Accuracy

Cited by in corpus (9)

• An adaptive BDF2 implicit time-stepping method for the phase field crystal model
• Mesh-robustness of an energy stable BDF2 scheme with variable steps for the Cahn-Hilliard model
• The variable-step L1 scheme preserving a compatible energy law for time-fractional Allen-Cahn equation
• Energy stability of variable-step L1-type schemes for time-fractional Cahn-Hilliard model
• Positive and free energy satisfying schemes for diffusion with interaction potentials
• Improved time integration for phase-field crystal models of solidification
• Spectral Deferred Correction Method for Landau-Brazovskii Model with Convex Splitting Technique
• Hybrid-PFC: coupling the phase-field crystal model and its amplitude-equation formulation
• Efficient numerical methods for computing the stationary states of phase field crystal models