4 research outputs found
Second Order Linear Energy Stable Schemes for Allen-Cahn Equations with Nonlocal Constraints
We present a set of linear, second order, unconditionally energy stable
schemes for the Allen-Cahn equation with nonlocal constraints that preserves
the total volume of each phase in a binary material system. The energy
quadratization strategy is employed to derive the energy stable semi-discrete
numerical algorithms in time. Solvability conditions are then established for
the linear systems resulting from the semi-discrete, linear schemes. The fully
discrete schemes are obtained afterwards by applying second order finite
difference methods on cell-centered grids in space. The performance of the
schemes are assessed against two benchmark numerical examples, in which
dynamics obtained using the volumepreserving Allen-Cahn equations with nonlocal
constraints is compared with those obtained using the classical Allen-Cahn as
well as the Cahn-Hilliard model, respectively, demonstrating slower dynamics
when volume constraints are imposed as well as their usefulness as alternatives
to the Cahn-Hilliard equation in describing phase evolutionary dynamics for
immiscible material systems while preserving the phase volumes. Some
performance enhancing, practical implementation methods for the linear energy
stable schemes are discussed in the end
Global constraints preserving SAV schemes for gradient flows
We develop several efficient numerical schemes which preserve exactly the
global constraints for constrained gradient flows. Our schemes are based on the
SAV approach combined with the Lagrangian multiplier approach. They are as
efficient as the SAV schemes for unconstrained gradient flows, i.e., only
require solving linear equations with constant coefficients at each time step
plus an additional nonlinear algebraic system which can be solved at negligible
cost, can be unconditionally energy stable, and preserve exactly the global
constraints for constrained gradient flows. Ample numerical results for
phase-field vesicle membrane and optimal partition models are presented to
validate the effectiveness and accuracy of the proposed numerical schemes
The generalized scalar auxiliary variable approach (G-SAV) for gradient flows
We establish a general framework for developing, efficient energy stable
numerical schemes for gradient flows and develop three classes of generalized
scalar auxiliary variable approaches (G-SAV). Numerical schemes based on the
G-SAV approaches are as efficient as the original SAV schemes
\cite{SXY19,cheng2018multiple} for gradient flows, i.e., only require solving
linear equations with constant coefficients at each time step, can be
unconditionally energy stable. But G-SAV approaches remove the definition
restriction that auxiliary variables can only be square root function. The
definition form of auxiliary variable is applicable to any reversible function
for G-SAV approaches . Ample numerical results for phase field models are
presented to validate the effectiveness and accuracy of the proposed G-SAV
numerical schemes
Linear Second Order Energy Stable Schemes of Phase Field Model with Nonlocal Constraints for Crystal Growth
We present a set of linear, second order, unconditionally energy stable
schemes for the Allen-Cahn model with a nonlocal constraint for crystal growth
that conserves the mass of each phase. Solvability conditions are established
for the linear systems resulting from the linear schemes. Convergence rates are
verified numerically. Dynamics obtained using the nonlocal Allen-Cahn model are
compared with the one obtained using the classic Allen-Cahn model as well as
the Cahn-Hilliard model, demonstrating slower dynamics than that of the
Allen-Cahn model but faster dynamics than that of the Cahn-Hillard model. Thus,
the nonlocal Allen-Cahn model can be an alternative to the Cahn-Hilliard model
in simulating crystal growth. Two Benchmark examples are presented to
illustrate the prediction made with the nonlocal Allen-Cahn model in comparison
to those made with the Allen-Cahn model and the Cahn- Hillard model.Comment: arXiv admin note: substantial text overlap with arXiv:1810.0531