We present and analyze a new second-order finite difference scheme for the Macromolecular Microsphere Composite hydrogel, Time-Dependent Ginzburg-Landau (MMC-TDGL) equation, a Cahn-Hilliard equation with Flory-Huggins-deGennes energy potential. This numerical scheme with unconditional energy stability is based on the Backward Differentiation Formula (BDF) method time derivation combining with Douglas-Dupont regularization term. In addition, we present a point-wise bound of the numerical solution for the proposed scheme in the theoretical level. For the convergent analysis, we treat three nonlinear logarithmic terms as a whole and deal with all logarithmic terms directly by using the property that the nonlinear error inner product is always non-negative. Moreover, we present the detailed convergent analysis in $\ell^\infty (0,T; H_h^{-1}) \cap \ell^2 (0,T; H_h^1)$ norm. At last, we use the local Newton approximation and multigrid method to solve the nonlinear numerical scheme, and various numerical results are presented, including the numerical convergence test, positivity-preserving property test, spinodal decomposition, energy dissipation and mass conservation properties.
Recently, the error analysis of BDF$k$ $(1\leqslant k\leqslant5)$ SAV (scalar auxiliary variable) schemes are given in \cite{Huangg:20} for the classical Allen-Cahn equation. However, it remains unavailable for BDF$6$ SAV schemes. In this paper, we construct and analyze BDF$6$ SAV schemes for the time-fractional dissipative systems. We carry out a rigorous error analysis for the time-fractional Allen-Cahn equation, which also fills up a gap for the classical case. Finally, numerical experiment is shown to illustrate the effectiveness of the presented methods. As far as we know, this is the fir…
In this work, we investigate the two-step backward differentiation formula (BDF2) with nonuniform grids for the Allen-Cahn equation. We show that the nonuniform BDF2 scheme is energy stable under the time-step ratio restriction $r_k:=τ_k/τ_{k-1}<(3+\sqrt{17})/2\approx3.561.$ Moreover, by developing a novel kernel recombination and complementary technique, we show, for the first time, the discrete maximum principle of BDF2 scheme under the time-step ratio restriction $r_k<1+\sqrt{2}\approx 2.414$ and a practical time step constraint. The second-order rate of convergence in the maximum norm is a…
The space fractional Cahn-Hilliard phase-field model is more adequate and accurate in the description of the formation and phase change mechanism than the classical Cahn-Hilliard model. In this article, we propose a temporal second-order energy stable scheme for the space fractional Cahn-Hilliard model. The scheme is based on the second-order backward differentiation formula in time and a finite difference method in space. Energy stability and convergence of the scheme are analyzed, and the optimal convergence orders in time and space are illustrated numerically. Note that the coefficient matr…
An adaptive implicit-explicit (IMEX) BDF2 scheme is investigated on generalized SAV approach for the Cahn-Hilliard equation by combining with Fourier spectral method in space. It is proved that the modified energy dissipation law is unconditionally preserved at discrete levels. Under a mild ratio restriction, i.e., \Ass{1}: $0<r_k:=τ_k/τ_{k-1}< r_{\max}\approx 4.8645$, we establish a rigorous error estimate in $H^1$-norm and achieve optimal second-order accuracy in time. The proof involves the tools of discrete orthogonal convolution (DOC) kernels and inequality zoom. It is worth noting that t…
We consider the $C^1$-Virtual Element Method (VEM) for the conforming numerical approximation of some variants of the Cahn-Hilliard equation on polygonal meshes. In particular, we focus on the discretization of the advective Cahn-Hilliard problem and the Cahn-Hilliard inpainting problem. We present the numerical approximation and several numerical results to assess the efficacy of the proposed methodology.
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.
The two-step backward differential formula (BDF2) with unequal time-steps is applied to construct an energy stable convex-splitting scheme for the Cahn-Hilliard model. We focus on the numerical influences of time-step variations by using the recent theoretical framework with the discrete orthogonal convolution kernels. Some novel discrete convolution embedding inequalities with respect to the orthogonal convolution kernels are developed such that a concise $L^2$ norm error estimate is established at the first time under an updated step-ratio restriction $0 <r_k:=τ_k/τ_{k-1}\leq r_{\mathrm{user…
In this paper, we review modified Cahn-Hilliard equation for image inpainting and explore the effect when the initial condition is uncertain. We study the statistical properties of the solution when the noise is present. The generalized polynomial chaos and the perturbation expansion are used to analyze the equation. Experimental results are attached for comparison of solution behavior.
This work addresses the problem of solving the Cahn-Hilliard equation numerically. For that we introduce an abstract formulation for Cahn-Hilliard type equations with dynamic boundary conditions, we conduct the spatial semidiscretization via finite elements and prove error bounds based on the technique of energy estimates. The variational formulation for Cahn-Hilliard/Cahn-Hilliard coupling, will apply to a larger abstract class of problems and is similar to the usual weak formulation of parabolic problems. In contrast to problems with non dynamic boundary conditions, the Hilbert spaces $L^2(Ω…
We propose a numerical approximation method for the Cahn-Hilliard equations that incorporates continuous data assimilation in order to achieve long time accuracy. The method uses a C$^0$ interior penalty spatial discretization of the fourth order Cahn-Hilliard equations, together with a backward Euler temporal discretization. We prove the method is long time stable and long time accurate, for arbitrarily inaccurate initial conditions, provided enough data measurements are incorporated into the simulation. Numerical experiments illustrate the effectiveness of the method on a benchmark test prob…
In this paper, we propose and analyze a linear second-order numerical method for solving the Allen-Cahn equation with a general mobility. The proposed fully-discrete scheme is carefully constructed based on the combination of first and second-order backward differentiation formulas with nonuniform time steps for temporal approximation and the central finite difference for spatial discretization. The discrete maximum bound principle is proved of the proposed scheme by using the kernel recombination technique under certain mild constraints on the time steps and the ratios of adjacent time step s…
A proof of convergence is given for bulk--surface finite element semi-discretisation of the Cahn--Hilliard equation with Cahn--Hilliard-type dynamic boundary conditions in a smooth domain. The semi-discretisation is studied in the weak formulation as a second order system. Optimal-order uniform-in-time error estimates are shown in the $L^2$ and $H^1$ norms. The error estimates are based on a consistency and stability analysis. The proof of stability is performed in an abstract framework, based on energy estimates exploiting the anti-symmetric structure of the second order system. Numerical exp…
We construct a divergence-free HDG scheme for the Cahn-Hilliard-Navier-Stokes phase field model. The scheme is robust in the convection-dominated regime, produce a globally divergence-free velocity approximation, and can be efficiently implemented via static condensation. Two numerical benchmark problems, namely the rising bubble problem, and the Rayleigh-Taylor instability problem are used to show the good performance of the proposed scheme.
We derive a posteriori error estimates for a fully discrete finite element approximation of the stochastic Cahn-Hilliard equation. The a posteriori bound is obtained by a splitting of the equation into a linear stochastic partial differential equation (SPDE) and a nonlinear random partial differential equation (RPDE). The resulting estimate is robust with respect to the interfacial width parameter and is computable since it involves the discrete principal eigenvalue of a linearized (stochastic) Cahn-Hilliard operator. Furthermore, the estimate is robust with respect to topological changes as w…
The Cahn--Hilliard equation is a widely used model that describes amongst others phase separation processes of binary mixtures or two-phase flows. In the recent years, different types of boundary conditions for the Cahn--Hilliard equation were proposed and analyzed. In this publication, we are concerned with the numerical treatment of a recent model which introduces an additional Cahn--Hilliard type equation on the boundary as closure for the Cahn--Hilliard equation in the domain [C. Liu, H. Wu, Arch. Ration. Mech. An., 2019]. By identifying a mapping between the phase-field parameter and the …
In this paper, we analyze a substructuring type algorithm for the Cahn-Hilliard (CH) equation. Being a nonlinear equation, it is of great importance to develop efficient numerical schemes for investigating the solution behaviour of the CH equation. We present the formulation of Neumann-Neumann (NN) method applied to the CH equation and investigate the convergence behaviour of the same in one and two spatial dimension for two subdomains. We illustrate the theoretical results by providing numerical example.
We construct efficient implicit-explicit BDF$k$ scalar auxiliary variable (SAV) schemes for general dissipative systems. We show that these schemes are unconditionally stable, and lead to a uniform bound of the numerical solution in the norm based on the principal linear operator in the energy. Based on this uniform bound, we carry out a rigorous error analysis for the $k$th-order $(k=1,2,3,4,5)$ SAV schemes in a unified form for a class of typical Allen-Cahn type and Cahn-Hilliard type equations. We also present numerical results confirming our theoretical convergence rates.
Numerical approximations of the Cahn-Hilliard and Allen-Cahn Equations with general nonlinear potential using the Invariant Energy Quadratization approach
University of South Carolina（南卡罗来纳大学）；Yantai University（烟台大学）
This paper deals with time stepping schemes for the Cahn--Hilliard equation with three different types of dynamic boundary conditions. The proposed schemes of first and second order are mass-conservative and energy-dissipative and -- as they are based on a formulation as a coupled system of partial differential equations -- allow different spatial discretizations in the bulk and on the boundary. The latter enables refinements on the boundary without an adaptation of the mesh in the interior of the domain. The resulting computational gain is illustrated in numerical experiments.
We study the numerical algorithm and error analysis for the Cahn-Hilliard equation with dynamic boundary conditions. A second-order in time, linear and energy stable scheme is proposed, which is an extension of the first-order stabilized approach. The corresponding energy stability and convergence analysis of the scheme are derived theoretically. Some numerical experiments are performed to verify the effectiveness and accuracy of the second-order numerical scheme, including numerical simulations under various initial conditions and energy potential functions, and comparisons with the literatur…
The mixed form of the Cahn-Hilliard equations is discretized by the hybridizable discontinuous Galerkin method. For any chemical energy density, existence and uniqueness of the numerical solution is obtained. The scheme is proved to be unconditionally stable. Convergence of the method is obtained by deriving a priori error estimates that are valid for the Ginzburg-Lindau chemical energy density and for convex domains. The paper also contains discrete functional tools, namely discrete Agmon and Gagliardo-Nirenberg inequalities, which are proved to be valid in the hybridizable discontinuous Gale…
Kernel-free Boundary Integral Methods for Allen-Cahn and Cahn-Hilliard Equations on Irregular Domains
School of Mathematical Sciences and Institute of Natural Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院和自然科学研究院）；School of Mathematical Sciences, MOE-LSC and Institute of Natural Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院、教育部拉格朗日计算科学中心和自然科学研究院）
We prove energy stability of a standard operator-splitting method for the Cahn-Hilliard equation. We establish uniform bound of Sobolev norms of the numerical solution and convergence of the splitting approximation. This is the first unconditional energy stability result for the operator-splitting method for the Cahn-Hilliard equation. Our analysis can be extended to many other models.
This paper tackles the approximation of surface diffusion flow using a Cahn--Hilliard-type model. We introduce and analyze a new second order variational phase field model which associates the classical Cahn--Hilliard energy with two degenerate mobilities. This association allows to gain an order of approximation of the sharp limit. In a second part, we propose some simple and efficient numerical schemes to approximate the solutions, and we provide numerical 2D and 3D experiments that illustrate the interest of our model in comparison with other Cahn--Hilliard models.
We consider a Strang-type second order operator-splitting discretization for the Cahn-Hilliard equation. We introduce a new theoretical framework and prove uniform energy stability of the numerical solution and persistence of all higher Sobolev norms. This is the first strong stability result for second order operator-splitting methods for the Cahn-Hilliard equation. In particular we settle several long-standing open issues in the work of Cheng, Kurganov, Qu and Tang \cite{Tang15}.
We consider the two-dimensional Cahn-Hilliard equation with logarithmic potentials and periodic boundary conditions. We employ the standard semi-implicit numerical scheme which treats the linear fourth-order dissipation term implicitly and the nonlinear term explicitly. Under natural constraints on the time step we prove strict phase separation and energy stability of the semi-implicit scheme. This appears to be the first rigorous result for the semi-implicit discretization of the Cahn-Hilliard equation with singular potentials.
We prove a weak rate of convergence of a fully discrete scheme for stochastic Cahn--Hilliard equation with additive noise, where the spectral Galerkin method is used in space and the backward Euler method is used in time. Compared with the Allen--Cahn type stochastic partial differential equation, the error analysis here is much more sophisticated due to the presence of the unbounded operator in front of the nonlinear term. To address such issues, a novel and direct approach has been exploited which does not rely on a Kolmogorov equation but on the integration by parts formula from Malliavin c…
Phase field models, in particular, the Allen-Cahn type and Cahn-Hilliard type equations, have been widely used to investigate interfacial dynamic problems. Designing accurate, efficient, and stable numerical algorithms for solving the phase field models has been an active field for decades. In this paper, we focus on using the deep neural network to design an automatic numerical solver for the Allen-Cahn and Cahn-Hilliard equations by proposing an improved physics informed neural network (PINN). Though the PINN has been embraced to investigate many differential equation problems, we find a dir…
Cahn-Hilliard models are central for describing the evolution of interfaces in phase separation processes and free boundary problems. In general, they have non-constant and often degenerate mobilities. However, in the latter case, the spontaneous appearance of points of vanishing mobility and their impact on the solution are not well understood. In this paper we develop a singular perturbation theory to identify a range of degeneracies for which the solution of the Cahn-Hilliard equation forms a singularity in infinite time. This analysis forms the basis for a rigorous sharp interface theory a…
