162 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
A new class of efficient and robust energy stable schemes for gradient flows
We propose a new numerical technique to deal with nonlinear terms in gradient
flows. By introducing a scalar auxiliary variable (SAV), we construct efficient
and robust energy stable schemes for a large class of gradient flows. The SAV
approach is not restricted to specific forms of the nonlinear part of the free
energy, and only requires to solve {\it decoupled} linear equations with {\it
constant coefficients}. We use this technique to deal with several challenging
applications which can not be easily handled by existing approaches, and
present convincing numerical results to show that our schemes are not only much
more efficient and easy to implement, but can also better capture the physical
properties in these models. Based on this SAV approach, we can construct
unconditionally second-order energy stable schemes; and we can easily construct
even third or fourth order BDF schemes, although not unconditionally stable,
which are very robust in practice. In particular, when coupled with an adaptive
time stepping strategy, the SAV approach can be extremely efficient and
accurate
Convergence Analysis of an Unconditionally Energy Stable Linear Crank-Nicolson Scheme for the Cahn-Hilliard Equation
Efficient and unconditionally stable high order time marching schemes are
very important but not easy to construct for nonlinear phase dynamics. In this
paper, we propose and analysis an efficient stabilized linear Crank-Nicolson
scheme for the Cahn-Hilliard equation with provable unconditional stability. In
this scheme the nonlinear bulk force are treated explicitly with two
second-order linear stabilization terms. The semi-discretized equation is a
linear elliptic system with constant coefficients, thus robust and efficient
solution procedures are guaranteed. Rigorous error analysis show that, when the
time step-size is small enough, the scheme is second order accurate in time
with aprefactor controlled by some lower degree polynomial of .
Here is the interface thickness parameter. Numerical results are
presented to verify the accuracy and efficiency of the scheme.Comment: 26 pages, 2 figures. arXiv admin note: substantial text overlap with
arXiv:1708.0976
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
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
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
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
The Phase Field Method for Geometric Moving Interfaces and Their Numerical Approximations
This paper surveys recent numerical advances in the phase field method for
geometric surface evolution and related geometric nonlinear partial
differential equations (PDEs). Instead of describing technical details of
various numerical methods and their analyses, the paper presents a holistic
overview about the main ideas of phase field modeling, its mathematical
foundation, and relationships between the phase field formalism and other
mathematical formalisms for geometric moving interface problems, as well as the
current state-of-the-art of numerical approximations of various phase field
models with an emphasis on discussing the main ideas of numerical analysis
techniques. The paper also reviews recent development on adaptive grid methods
and various applications of the phase field modeling and their numerical
methods in materials science, fluid mechanics, biology and image science.Comment: 66 page
A Minimization Method for The Double-Well Energy Functional
In this paper an iterative minimization method is proposed to approximate the
minimizer to the double-well energy functional arising in the phase-field
theory. The method is based on a quadratic functional posed over a nonempty
closed convex set and is shown to be unconditionally energy stable. By the
minimization approach, we also derive an variant of the first-order scheme for
the Allen-Cahn equation, which has been constructed in the context of Invariant
Energy Quadratization, and prove its unconditional energy stability
Numerical Methods for a System of Coupled Cahn-Hilliard Equations
In this work, we study a system of coupled Cahn-Hilliard equations describing
the phase separation of a copolymer and a homopolymer blend. The numerical
methods we propose are based on suitable combinations of existing schemes for
the single Cahn-Hilliard equation. As a verification for our approach, we
present some tests and a detailed description of the numerical solutions'
behaviour obtained by varying the values of the parameters
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