7 research outputs found
On Efficient Second Order Stabilized Semi-Implicit Schemes for the Cahn-Hilliard Phase-Field Equation
Efficient and energy stable high order time marching schemes are very
important but not easy to construct for the study of nonlinear phase dynamics.
In this paper, we propose and study two linearly stabilized second order
semi-implicit schemes for the Cahn-Hilliard phase-field equation. One uses
backward differentiation formula and the other uses Crank-Nicolson method to
discretize linear terms. In both schemes, the nonlinear bulk forces are treated
explicitly with two second-order stabilization terms. This treatment leads to
linear elliptic systems with constant coefficients, for which lots of robust
and efficient solvers are available. The discrete energy dissipation properties
are proved for both schemes. Rigorous error analysis is carried out to show
that, when the time step-size is small enough, second order accuracy in time is
obtained with a prefactor controlled by a fixed power of , where
is the characteristic interface thickness. Numerical results are
presented to verify the accuracy and efficiency of proposed schemes
A stabilized semi-implicit Fourier spectral method for nonlinear space-fractional reaction-diffusion equations
The reaction-diffusion model can generate a wide variety of spatial patterns,
which has been widely applied in chemistry, biology, and physics, even used to
explain self-regulated pattern formation in the developing animal embryo. In
this work, a second-order stabilized semi-implicit time-stepping Fourier
spectral method is presented for the reaction-diffusion systems of equations
with space described by the fractional Laplacian. We adopt the temporal-spatial
error splitting argument to illustrate that the proposed method is stable
without imposing the CFL condition, and we prove an optimal L2-error estimate.
We also analyze the linear stability of the stabilized semi-implicit method and
obtain a practical criterion to choose the time step size to guarantee the
stability of the semi-implicit method. Our approach is illustrated by solving
several problems of practical interest, including the fractional Allen-Cahn,
Gray-Scott and FitzHugh-Nagumo models, together with an analysis of the
properties of these systems in terms of the fractional power of the underlying
Laplacian operator, which are quite different from the patterns of the
corresponding integer-order model.Comment: 24 pages, 7 figure
Energy Stable Second Order Linear Schemes for the Allen-Cahn Phase-Field Equation
Phase-field model is a powerful mathematical tool to study the dynamics of
interface and morphology changes in fluid mechanics and material sciences.
However, numerically solving a phase field model for a real problem is a
challenge task due to the non-convexity of the bulk energy and the small
interface thickness parameter in the equation. In this paper, we propose two
stabilized second order semi-implicit linear schemes for the Allen-Cahn
phase-field equation based on backward differentiation formula and
Crank-Nicolson method, respectively. In both schemes, the nonlinear bulk force
is treated explicitly with two second-order stabilization terms, which make the
schemes unconditional energy stable and numerically efficient. By using a known
result of the spectrum estimate of the linearized Allen-Cahn operator and some
regularity estimate of the exact solution, we obtain an optimal second order
convergence in time with a prefactor depending on the inverse of the
characteristic interface thickness only in some lower polynomial order. Both
2-dimensional and 3-dimensional numerical results are presented to verify the
accuracy and efficiency of proposed schemes.Comment: keywords: energy stable, stabilized semi-implicit scheme, second
order scheme, error estimate. related work arXiv:1708.09763, arXiv:1710.0360
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
Second-order Decoupled Energy-stable Schemes for Cahn-Hilliard-Navier-Stokes equations
The Cahn-Hilliard-Navier-Stokes (CHNS) equations represent the fundamental
building blocks of hydrodynamic phase-field models for multiphase fluid flow
dynamics. Due to the coupling between the Navier-Stokes equation and the
Cahn-Hilliard equation, the CHNS system is non-trivial to solve numerically.
Traditionally, a numerical extrapolation for the coupling terms is used.
However, such brute-force extrapolation usually destroys the intrinsic
thermodynamic structures of this CHNS system. This paper proposes a new
strategy to reformulate the CHNS system into a constraint gradient flow
formation. Under the new formulation, the reversible and irreversible
structures are clearly revealed. This guides us to propose operator splitting
schemes. The operator splitting schemes have several advantageous properties.
First of all, the proposed schemes lead to several decoupled systems in smaller
sizes to be solved at each time marching step. This significantly reduces
computational costs. Secondly, the proposed schemes still guarantee the
thermodynamic laws of the CHNS system at the discrete level. It ensures the
thermodynamic laws, accuracy, and stability for the numerical solutions. In
addition, unlike the recently populated IEQ or SAV approach using auxiliary
variables, our resulting energy laws are formulated in the original variables.
Our proposed framework lays out a foundation to design decoupled and energy
stable numerical algorithms for hydrodynamic phase-field models. Furthermore,
given different splitting steps, various numerical algorithms can be obtained,
making this framework rather general. The proposed numerical algorithms are
implemented. Their second-order accuracy in time is verified numerically. Some
numerical examples and benchmark problems are calculated to verify the
effectiveness of the proposed 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
An Energy Stable Linear Diffusive Crank-Nicolson Scheme for the Cahn-Hilliard Gradient Flow
We propose and analyze a linearly stabilized semi-implicit diffusive
Crank--Nicolson scheme for the Cahn--Hilliard gradient flow. In this scheme,
the nonlinear bulk force is treated explicitly with two second-order
stabilization terms. This treatment leads to linear elliptic system with
constant coefficients and provable discrete energy dissipation. Rigorous error
analysis is carried out for the fully discrete scheme. When the time step-size
and the space step-size are small enough, second order accuracy in time is
obtained with a prefactor controlled by some lower degree polynomial of
. {Here is the thickness of the interface}.
Numerical results together with an adaptive time stepping are presented to
verify the accuracy and efficiency of the proposed scheme.Comment: arXiv admin note: text overlap with arXiv:1710.03604,
arXiv:1708.0976