Nonlocal Allen Cahn Equation Research Articles

Quantitative convergence of the nonlocal Allen–Cahn equation to volume-preserving mean curvature flow

AbstractWe prove a quantitative convergence result of the nonlocal Allen–Cahn equation to volume-preserving mean curvature flow. The proof uses gradient flow calibrations and the relative entropy method, which has been used in the recent literature to prove weak–strong uniqueness results for mean curvature flow and convergence of the Allen–Cahn equation. A crucial difference in this work is a new notion of gradient flow calibrations. We add a tangential component to the velocity field in order to prove the Gronwall estimate for the relative energy. This allows us to derive the optimal convergence rate without having to show the closeness of the Lagrange-multipliers.

An effective numerical method for the vector-valued nonlocal Allen–Cahn equation

Nonlocal Allen–Cahn model and their numerical schemes have received great attention in the literature as nonlocal model becomes popular in various fields. Our main idea in this work is to consider the vector-valued nonlocal Allen–Cahn model, which is a coupled system of nonlinear partial differential equations. Then, with the help of the operator splitting method and finite difference method, a fully-decoupled and energy stable numerical scheme for the vector-valued nonlocal Allen–Cahn model is proposed. Furthermore, we prove the modified energy dissipation law is unconditionally guaranteed and it is closely related to classical energy up to O(τ). Finally, numerical examples are carried out to verify the efficiency and accuracy of the proposed scheme.

Maximum bound principle preserving linear schemes for nonlocal Allen–Cahn equation based on the stabilized exponential-SAV approach

Maximum bound principle preserving linear schemes for nonlocal Allen–Cahn equation based on the stabilized exponential-SAV approach

High-order unconditionally maximum-principle-preserving parametric integrating factor Runge-Kutta schemes for the nonlocal Allen-Cahn equation

We utilize the second-order quadrature-based finite difference method and the high-order parametric integrating factor Runge-Kutta (pIFRK) integrators to construct efficient and accurate schemes for solving the nonlocal Allen-Cahn equation. These schemes preserve the maximum principle for any time-step, and exhibit up to fourth-order accuracy in the temporal direction. We establish a rigorous error estimate and an asymptotic compatibility analysis for the pIFRK schemes. Numerical experiments demonstrate the accuracy and structure-preserving property of the proposed schemes, verify their asymptotic compatibility, and investigate the discontinuity of the nonlocal Allen-Cahn equation under certain conditions.

A new high-order maximum-principle-preserving explicit Runge–Kutta method for the nonlocal Allen–Cahn equation

We develop the high-order maximum principle preserving schemes based on combining the scalar variable with the explicit strong stability preserving Runge–Kutta (eSSPRK) methods, which can be applied to a class of gradient flows. In this work, we use the methods to deal with the nonlocal Allen–Cahn equation. There are two remarkable properties of the scheme. On the one hand, by using the introduced scalar variable to control the nonlinear term, and adopting the eSSPRK method to temporal discretization, we only need to solve the linear equations with constant coefficients at each time level. On the other hand, we introduce the first-order approximation of the energy balance equation instead of the dynamic equation for the scalar variable, and we demonstrate that the proposed schemes are unconditionally energy stable. In particular, the discrete schemes are verified to preserve the maximum bound principle under the time-step restriction. Moreover, error estimates and the asymptotic compatibility of the proposed methods are proved, respectively. Numerical examples are performed to illustrate the efficiency and accuracy of the numerical schemes.

A conservative phase‐field lattice Boltzmann method for incompressible two‐phase flows

AbstractIn this work, a conservative phase‐field lattice Boltzmann method (LBM) is proposed for incompressible two‐phase flows. In this method, two LB models are adopted to solve the nonlocal Allen–Cahn equation (ACE) and the Navier–Stokes equations, respectively. Unlike the previous models for the nonlocal ACE with a space‐time‐dependent Lagrange multiplier, a simple multiple‐relaxation time LB model is designed for the conservative and nonlocal ACE with a curvature‐dependent Lagrange multiplier, which can eliminate the problem of inherent coarsening process. Through the Chapman–Enskog analysis, the present LB model can correctly recover the ACE, and the macroscopic order parameter used to label different phases can be calculated explicitly. Finally, the present LBM is validated by two tests, that is, the horizontal translation of two circular interfaces and a bubble rising under gravity, and the results show that the present LBM has a superior performance in capturing the large topological changes and small features of interfaces.

Efficient, linear and fast numerical algorithm for the volume conserved nonlocal Allen-Cahn equation

In this article, for the conserved nonlocal Allen-Cahn equation, we construct a set of efficient, linear, unconditionally energy stable numerical schemes based on the Invariant Energy Quadratization approach. We not only show that the proposed schemes satisfy the law of energy dissipation unconditionally, but also derive the error estimates rigorously. Meanwhile, to reduce the computational cost and memory requirement caused by the nonlocal term, a fast implementation process based on the FFT technique is developed. Some numerical simulations are carried out to demonstrate the accuracy and stability of the schemes as well.

Local hybrid Allen-Cahn model in phase-field lattice Boltzmann method for incompressible two-phase flow.

For simulating incompressible two-phase fluid flows, several phase-field lattice Boltzmann (LB) methods based on the local Allen-Cahn (AC) equation have been intensively proposed in recent years. We present a local hybrid AC model for the phase-field LB method. In the proposed model, the local and nonlocal AC equations are linearly combined using a local weight assigned in the interface or bulk phase regions individually. Five numerical problems, namely diagonal translation, Zalesak's disk rotation, static bubble, two bubbles of different radii, and Rayleigh-Taylor instability, are simulated for validation. The numerical results agree well with the analytical solutions or available previous results. Additionally, the numerical dispersion and the coarsening phenomenon are considerably suppressed in the proposed model. Finally, the performance of the proposed model is validated by conducting a drainage simulation in porous media and compared with the global hybrid AC model.

An effective operator splitting scheme for two-dimensional conservative nonlocal Allen–Cahn equation

An effective operator splitting scheme for solving the nonlocal Allen–Cahn equation with a Lagrange multiplier is studied. Firstly, based on the operator splitting method, the original equation is discretized into a nonlinear equation, a nonlocal equation and a Lagrange multiplier equation, respectively. Then, the nonlinear equation is solved analytically, the nonlocal equation is discretized using Crank–Nicolson format, and the Lagrange multiplier equation is solved explicitly. The mass conservation and convergence analysis of the numerical algorithm are analyzed theoretically. Numerical experiments are presented to confirm the validity of the proposed method, including the convergence, energy decline and mass conservation.

The high-order maximum-principle-preserving integrating factor Runge-Kutta methods for nonlocal Allen-Cahn equation

We extend the explicit integrating factor Runge-Kutta methods coupled with non-decreasing abscissas (eIFRK+) to the nonlocal Allen-Cahn (NAC) equation. We further propose the new three-stage third-order and four-stage fourth-order eIFRK+ schemes based on the classic RK method, which can be used for a class of local and nonlocal models. In this paper, the method is mainly applied to study the NAC equation. Under a large time-step constraint, the high-order eIFRK+ schemes are demonstrated to preserve maximum bound principle, which is a crucial physical property for the NAC models. Then, the optimal error estimates in L∞(0,T;Ω)-norm are established and the asymptotic compatibility of the proposed schemes are validated. Numerical experiments are carried out to verify our theoretical results and illustrate the effectiveness of the fully discrete schemes. Moreover, by the aid of numerical simulation, we attempt to declare that the eIFRK+ schemes are energy stable under the weak time-step restriction.

Fast algorithm for nonlocal Allen–Cahn equation with scalar auxiliary variable approach

Numerical analysis is presented for the nonlocal Allen–Cahn equation, which contains spatial nonlocal operator and time-fractional derivative. By employing the spatial quadrature-based finite difference method and the nonuniform L1 formula jointed with the scalar auxiliary variable (SAV) approach in temporal discretization, a nonuniform numerical scheme is established. The nonlinear solver can be transformed into linear one effectively due to the SAV approach. And, the proposed scheme is proven to be energy stable by use of the positive definiteness of the kernel function. Moreover, the fast algorithm based on the nonuniform L1 formula is applied in the numerical example to improving computational efficiency. Finally, the numerical results demonstrate the temporal convergence of numerical scheme, energy property, comparisons with the nonlocal cases and local cases and maximum principle of the numerical solution.

Second order linear thermodynamically consistent approximations to nonlocal phase field porous media models

We present a weakly compressible, thermodynamically consistent, hydrodynamical model for the binary fluid flow in porous media using the one-fluid multiple component formulation, consisting of a force balance equation with a weak inertia, weakly perturbed continuity equation, and a volume-preserving, nonlocal Allen–Cahn equation for the phase field variable. This model is called the nonlocal Allen–Cahn-Extended-Darcy (NACED) model. In the incompressible and/or the inertialess limit, it reduces to two limiting models that we focus on in this study: the incompressible, nonlocal Allen–Cahn-Extended-Darcy (NACED) model and nonlocal Allen–Cahn–Darcy (NACD) model, respectively. The weakly compressible model provides a pathway for us to develop thermodynamically consistent numerical algorithms for the incompressible models. Guided by the energy variational formulation of the models, we use the energy quadratization method in time and finite difference method in space on staggered grids to derive second-order, linear, coupled and decoupled, energy dissipation rate preserving schemes for the incompressible models. The decoupled schemes for NACED model are obtained by exploiting the intrinsic relation between the incompressible and weakly compressible model. We then prove that the linear systems resulted from the linear schemes are uniquely solvable. Mesh refinement tests are conducted to confirm the convergence rates and benchmark examples of binary fluid flow motion in porous media with and without gravity are presented to showcase the schemes’ accuracy and usefulness.

Stability for stationary solutions of a nonlocal Allen-Cahn equation

We consider the dynamics of a nonlocal Allen-Cahn equation with Neumann boundary conditions on an interval. Our previous papers [2,3] obtained the global bifurcation diagram of stationary solutions, which includes the secondary bifurcation from the odd symmetric solution due to the symmetric breaking effect. This paper derives the stability/instability of all symmetric solutions and instability of a part of asymmetric solutions. To do so, we use the exact representation of symmetric solutions and show the distribution of eigenvalues of the linearized eigenvalue problem around these solutions. And we show the instability of asymmetric solutions by the SLEP method. Finally, our results with respect to stability are supported by some numerical simulations.

An Explicit Hybrid Method for the Nonlocal Allen–Cahn Equation

We extend the explicit hybrid numerical method for solving the Allen–Cahn (AC) equation to the scheme for the nonlocal AC equation with isotropically symmetric interfacial energy. The proposed method combines the previous explicit hybrid method with a space-time dependent Lagrange multiplier which enforces conservation of mass. We perform numerical tests for the area-preserving mean curvature flow, which is the basic property of the nonlocal AC equation. The numerical results show good agreement with the theoretical solutions. Furthermore, to demonstrate the usefulness of the proposed method, we perform a cell growth simulation in a complex domain. Because the proposed numerical scheme is explicit, it is remarkably simple to implement the numerical solution algorithm on complex discrete domains.

Representation formulas of solutions and bifurcation sheets to a nonlocal Allen-Cahn equation

We are interested in the Neumann problem of a 1D stationary Allen-Cahn equation with a nonlocal term. In our previous papers [ 4 ] and [ 5 ], we obtained a global bifurcation branch, and showed the existence and uniqueness of secondary bifurcation point. At this point, asymmetric solutions bifurcate from a branch of odd-symmetric solutions. In this paper, we give representation formulas of all solutions on the secondary bifurcation branch, and a bifurcation sheet which consists of bifurcation curves with heights.

Efficient, non-iterative, and second-order accurate numerical algorithms for the anisotropic Allen–Cahn Equation with precise nonlocal mass conservation

We propose two efficient, non-iterative, and second-order accurate algorithms to solving the anisotropic Allen–Cahn equation with the nonlocal mass conservation. The first scheme is the stabilized-SAV approach which combines the SAV approach with the stabilization technique, in which three linear stabilization terms are added to enhance the stability and keep the required accuracy while using large time steps. The scheme not only can conserve the mass precisely but also is very easy-to-implement and non-iterative where one only needs to solve three decoupled fourth-order biharmonic equations with constant coefficients at each time step. We further prove the scheme is unconditionally energy stable rigorously. The second scheme is based on the linear stabilization approach where all nonlinear terms are treated in an explicit way, therefore the scheme is quite efficient and stable that allows for large time steps in computations. For both schemes, we present a number of 2D and 3D numerical simulations to show stability and accuracy. • The anisotropic Cahn-Hilliard model is reformulated by using nonlocal Allen-Cahn equation. • Two efficient, non-iterative, and second-order algorithms are developed to preserve the mass. • Three linear stabilization terms are added to enhance the stability that allows large time step. • Numerous 2D and 3D numerical simulations to demonstrate the stability and accuracy.

Hybrid Allen-Cahn-based lattice Boltzmann model for incompressible two-phase flows: The reduction of numerical dispersion.

In this work a phase field-lattice Boltzmann model for incompressible two-phase flows is presented. In this model, the interface tracking equation is a linear combination of the local and nonlocal Allen-Cahn equations. We also propose a multiple-relaxation-time lattice Boltzmann model for solving the hybrid Allen-Cahn equation. The second-order convergence rate of the present model in space is validated by simulating the diagonal translation of circular interface. Three other numerical tests, including static bubble immersed in another fluid, bubble rising under gravity, and droplet splashing on a thin liquid film, are simulated to verify the performance of the present model in reducing the numerical dispersion. The numerical results indicate that the order parameter fluctuation can be reduced by one order of magnitude in bulk region.

A stable and structure-preserving scheme for a non-local Allen–Cahn equation

We propose a stable and structure-preserving finite difference scheme for a non-local Allen–Cahn equation which describes a process of phase separation in a binary mixture. The proposed scheme inherits characteristic properties, the conservation of mass and the decrease of the global energy from the equation. We show the stability and unique existence of the solution of the scheme. We also prove the error estimate for the scheme. Numerical experiments demonstrate the effectiveness of the proposed scheme.

A comparative study of local and nonlocal Allen-Cahn equations with mass conservation

The local and nonlocal Allen-Cahn equations (ACEs) have received increasing attention in the study of the complicated interfacial problems. In this paper, we conduct a comparison between local and nonlocal ACEs with the property of mass conservation in the framework of lattice Boltzmann (LB) method. To this end, we first propose two simple multiple-relaxation-time LB models for local and nonlocal ACEs, and through the Chapman-Enskog expansion, the local and nonlocal ACEs can be recovered correctly from the developed LB models. Then we test these two LB models with several examples, and the numerical results show that the developed LB models are accurate and also have a second-order convergence rate in space. Finally, a comparison between the local and nonlocal ACEs is also performed in terms of mass conservation, stability and accuracy. The results show that both local and nonlocal ACEs can preserve mass conservation of system and each phase. And additionally, it is also found that the local ACE is more accurate than nonlocal ACE in capturing the interface profile, but the latter is more stable than the former.

Global solution branches for a nonlocal Allen–Cahn equation

We consider the Neumann problem of a 1D stationary Allen–Cahn equation with nonlocal term. Our previous paper [4] obtained a local branch of asymmetric solutions which bifurcates from a point on the branch of odd-symmetric solutions. This paper derives the global behavior of the branch of asymmetric solutions, and moreover, determines the set of all solutions to the nonlocal Allen–Cahn equation. Our proof is based on a level set analysis for an integral map associated with the nonlocal term.