Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends

Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method

Summary
We present two accurate and efficient numerical schemes for a phase field dendritic crystal growth model, which is derived from the variation of a free-energy functional, consisting of a temperature dependent bulk potential and a conformational entropy with a gradient-dependent anisotropic coefficient. We introduce a novel Invariant Energy Quadratization approach to transform the free-energy functional into a quadratic form by introducing new variables to substitute the nonlinear transformations. Based on the reformulated equivalent governing system, we develop a first and a second order semi-discretized scheme in time for the system, in which all nonlinear terms are treated semi-explicitly. The resulting semi-discretized equations consist of a linear elliptic equation system at each time step, where the coefficient matrix operator is positive definite and thus, the semi-discrete system can be solved efficiently. We further prove that the proposed schemes are unconditionally energy stable. Convergence test together with 2D and 3D numerical simulations for dendritic crystal growth are presented after the semi-discrete schemes are fully discretized in space using the finite difference method to demonstrate the stability and the accuracy of the proposed schemes. Copyright © 2016 John Wiley & Sons, Ltd.

Numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach

In this paper we present a second order accurate (in time) energy stable numerical scheme for the Cahn-Hilliard (CH) equation, with a mixed finite element approximation in space. Instead of the standard second order Crank-Nicolson methodology, we apply the implicit backward differentiation formula (BDF) concept to derive second order temporal accuracy, but modified so that the concave diffusion term is treated explicitly. This explicit treatment for the concave part of the chemical potential ensures the unique solvability of the scheme without sacrificing energy stability. An additional term Aτ∆(uk+1−uk) is added, which represents a second order Douglas-Dupont-type regularization, and a careful calculation shows that energy stability is guaranteed, provided the mild condition A ≥ 1 16 is enforced. In turn, a uniform in time H1 bound of the numerical solution becomes available. As a result, we are able to establish an l∞(0, T ;L2) convergence analysis for the proposed fully discrete scheme, with full O(τ2+h2) accuracy. This convergence turns out to be unconditional; no scaling law is needed between the time step size τ and the spatial grid size h. A few numerical experiments are presented to conclude the article.

A Second-Order Energy Stable BDF Numerical Scheme for the Cahn-Hilliard Equation

In this paper we devise and analyze an unconditionally stable, second-order-in-time numerical scheme for the Cahn-Hilliard equation in two and three space dimensions. We prove that our two-step scheme is unconditionally energy stable and unconditionally uniquely solvable. Furthermore, we show that the discrete phase variable is bounded in $L^\infty (0,T;L^\infty)$ and the discrete chemical potential is bounded in $L^\infty (0,T;L^2)$, for any time and space step sizes, in two and three dimensions, and for any finite final time $T$. We subsequently prove that these variables converge with optimal rates in the appropriate energy norms in both two and three dimensions. We include in this work a detailed analysis of the initialization of the two-step scheme.

Stability and convergence of a second-order mixed finite element method for the Cahn–Hilliard equation

In this paper, we present a novel second order in time mixed finite element scheme for the Cahn–Hilliard–Navier–Stokes equations with matched densities. The scheme combines a standard second order Crank–Nicolson method for the Navier–Stokes equations and a modification to the Crank–Nicolson method for the Cahn–Hilliard equation. In particular, a second order Adams-Bashforth extrapolation and a trapezoidal rule are included to help preserve the energy stability natural to the Cahn–Hilliard equation. We show that our scheme is unconditionally energy stable with respect to a modification of the continuous free energy of the PDE system. Specifically, the discrete phase variable is shown to be bounded in $$\ell ^\infty \left( 0,T;L^\infty \right) $$
 and the discrete chemical potential bounded in $$\ell ^\infty \left( 0,T;L^2\right) $$
 , for any time and space step sizes, in two and three dimensions, and for any finite final time T. We subsequently prove that these variables along with the fluid velocity converge with optimal rates in the appropriate energy norms in both two and three dimensions.

Convergence analysis and error estimates for a second order accurate finite element method for the Cahn–Hilliard–Navier–Stokes system

In this paper we devise and analyze a mixed finite element method for a modified Cahn--Hilliard equation coupled with a nonsteady Darcy--Stokes flow that models phase separation and coupled fluid flow in immiscible binary fluids and diblock copolymer melts. The time discretization is based on a convex splitting of the energy of the equation. We prove that our scheme is unconditionally energy stable with respect to a spatially discrete analogue of the continuous free energy of the system and unconditionally uniquely solvable. We prove that the discrete phase variable is bounded in $L^\infty \left(0,T;L^\infty\right)$ and the discrete chemical potential is bounded in $L^\infty \left(0,T;L^2\right)$, for any time and space step sizes, in two and three dimensions, and for any finite final time $T$. We subsequently prove that these variables converge with optimal rates in the appropriate energy norms in both two and three dimensions.

Analysis of a Mixed Finite Element Method for a Cahn--Hilliard--Darcy--Stokes System

In this paper we devise and analyze a mixed finite element method for a modified Cahn-Hilliard equation coupled with a non-steady Darcy-Stokes flow that models phase separation and coupled fluid flow in immiscible binary fluids and diblock copolymer melts The time discretization is based on a convex splitting of the energy of the equation We prove that our scheme is unconditionally energy stable with respect to a spatially discrete analogue of the continuous free energy of the system and unconditionally uniquely solvable We prove that the phase variable is bounded in $L^\infty \left(0,T,L^\infty\right)$ and the chemical potential is bounded in $L^\infty \left(0,T,L^2\right)$ absolutely unconditionally in two and three dimensions, for any finite final time $T$ We subsequently prove that these variables converge with optimal rates in the appropriate energy norms in both two and three dimensions

Analysis of a Mixed Finite Element Method for a Cahn-Hilliard-Darcy-Stokes System

In this paper, we present a novel second order in time mixed finite element scheme for the Cahn-Hilliard-Navier-Stokes equations with matched densities. The scheme combines a standard second order Crank-Nicholson method for the Navier-Stokes equations and a modification to the Crank-Nicholson method for the Cahn-Hilliard equation. In particular, a second order Adams-Bashforth extrapolation and a trapezoidal rule are included to help preserve the energy stability natural to the Cahn-Hilliard equation. We show that our scheme is unconditionally energy stable with respect to a modification of the continuous free energy of the PDE system. Specifically, the discrete phase variable is shown to be bounded in $\ell^\infty \left(0,T;L^\infty\right)$ and the discrete chemical potential bounded in $\ell^\infty \left(0,T;L^2\right)$, for any time and space step sizes, in two and three dimensions, and for any finite final time $T$. We subsequently prove that these variables along with the fluid velocity converge with optimal rates in the appropriate energy norms in both two and three dimensions.

Amanda E. Diegel

Papers

Stability and convergence of a second-order mixed finite element method for the Cahn–Hilliard equation

Convergence analysis and error estimates for a second order accurate finite element method for the Cahn–Hilliard–Navier–Stokes system

Analysis of a Mixed Finite Element Method for a Cahn--Hilliard--Darcy--Stokes System

Analysis of a Mixed Finite Element Method for a Cahn-Hilliard-Darcy-Stokes System

Convergence Analysis and Error Estimates for a Second Order Accurate Finite Element Method for the Cahn-Hilliard-Navier-Stokes System