Multigrid
Conservative multigrid methods for Cahn-Hilliard fluids
Journal of Computational Physics
On large time-stepping methods for the Cahn--Hilliard equation
Applied Numerical Mathematics
An efficient algorithm for solving the phase field crystal model
Journal of Computational Physics
Journal of Computational Physics
SIAM Journal on Numerical Analysis
Journal of Computational Physics
Journal of Scientific Computing
Journal of Computational Physics
SIAM Journal on Numerical Analysis
A gradient stable scheme for a phase field model for the moving contact line problem
Journal of Computational Physics
An adaptive multigrid algorithm for simulating solid tumor growth using mixture models
Mathematical and Computer Modelling: An International Journal
SIAM Journal on Numerical Analysis
On linear schemes for a Cahn-Hilliard diffuse interface model
Journal of Computational Physics
An adaptive time-stepping strategy for solving the phase field crystal model
Journal of Computational Physics
Journal of Computational Physics
High accuracy solutions to energy gradient flows from material science models
Journal of Computational Physics
Accurate, efficient, and (iso)geometrically flexible collocation methods for phase-field models
Journal of Computational Physics
Hi-index | 31.49 |
In this paper we present and compare two unconditionally energy stable finite-difference schemes for the phase field crystal equation. The first is a one-step scheme based on a convex splitting of a discrete energy by Wise et al. [S.M. Wise, C. Wang, J.S. Lowengrub, An energy stable and convergent finite-difference scheme for the phase field crystal equation, SIAM J. Numer. Anal., in press]. In this scheme, which is first order in time and second order in space, the discrete energy is non-increasing for any time step. The second scheme we consider is a new, fully second-order two-step algorithm. In the new scheme, the discrete energy is bounded by its initial value for any time step. In both methods, the equations at the implicit time level are nonlinear but represent the gradients of strictly convex functions and are thus uniquely solvable, regardless of time step-size. We solve the nonlinear equations using an efficient nonlinear multigrid method. Numerical simulations are presented and confirm the stability, efficiency and accuracy of the schemes.