A positivity-preserving second-order BDF scheme for the Cahn-Hilliard equation with variable interfacial parameters

Abstract

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.

arXiv commentarXiv admin note: text overlap with arXiv:1712.03225

References in corpus (4)

• A second order in time, uniquely solvable, unconditionally stable numerical scheme for Cahn-Hilliard-Navier-Stokes equation
• Energy Stable and Efficient Finite-Difference Nonlinear Multigrid Schemes for the Modified Phase Field Crystal Equation
• Preconditioned Steepest Descent Methods for some Nonlinear Elliptic Equations Involving p-Laplacian Terms
• An unconditionally energy stable finite difference scheme for a stochastic Cahn-Hilliard equation

Cited by in corpus (10)

• A positivity-preserving, energy stable scheme for a Ternary Cahn-Hilliard system with the singular interfacial parameters
• A structure-preserving, operator splitting scheme for reaction-diffusion equations with detailed balance
• A highly efficient and accurate exponential semi-implicit scalar auxiliary variable (ESI-SAV) approach for dissipative system
• A Positive and Energy Stable Numerical Scheme for the Poisson-Nernst-Planck-Cahn-Hilliard Equations with Steric Interactions
• A positivity-preserving and convergent numerical scheme for the binary fluid-surfactant system
• A Uniquely Solvable, Positivity-Preserving and Unconditionally Energy Stable Numerical Scheme for the Functionalized Cahn-Hilliard Equation with Logarithmic Potential
• A positivity-preserving, energy stable and convergent numerical scheme for the Poisson-Nernst-Planck system
• Structure-preserving, energy stable numerical schemes for a liquid thin film coarsening model
• An energy stable finite element scheme for the three-component Cahn-Hilliard-type model for macromolecular microsphere composite hydrogels
• A Thermodynamically-Consistent Phase Field Crystal Model of Solidification with Heat Flux