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 $1/\varepsilon$. Here $\varepsilon$ 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 $1/\varepsilon$, where $\varepsilon$ 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