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A positivity-preserving, second-order energy stable and convergent numerical scheme for a ternary system of macromolecular microsphere composite hydrogels

Lixiu Dong, Cheng Wang, Zhengru Zhang

TL;DR

The authors develop a second-order, positivity-preserving, energy-stable finite difference scheme for a ternary MMC Cahn-Hilliard system governed by a Flory-Huggins–deGennes free energy. By combining a convex-concave energy decomposition with a BDF2 time discretization and Douglas-Dupont regularization, they ensure unique solvability, mass conservation, and unconditional energy dissipation, while maintaining pointwise positivity for $(\phi_1,\phi_2)$ and their sum. They establish a rigorous convergence theory, including higher-order consistency, rough and refined error estimates, yielding second-order convergence in $L^2$ and $L^{\infty}$ norms under a linear refinement between time step and mesh size, and even higher-order $L^2$ convergence under the enhanced consistency framework. Numerical experiments corroborate the theoretical properties, demonstrating mass conservation, energy decay, positivity, and the expected second-order accuracy, validating the scheme's reliability for simulating MMC hydrogels.

Abstract

A second order accurate numerical scheme is proposed and analyzed for the periodic three-component Macromolecular Microsphere Composite(MMC) hydrogels system, a ternary Cahn-Hilliard system with a Flory-Huggins-deGennes free energy potential. This numerical scheme with energy stability is based on the Backward Differentiation Formula(BDF) method in time derivation combining with Douglas-Dupont regularization term, combined the finite difference method in space. We provide a theoretical justification of positivity-preserving property for all the singular terms, i.e., not only the two phase variables are always between $0$ and $1$, but also the sum of the two phase variables is between $0$ and $1$, at a point-wise level. In addition, an optimal rate convergence analysis is provided in this paper, in which a higher order asymptotic expansion of the numerical solution, the rough error estimate and refined error estimate techniques have to be included to accomplish such an analysis. This paper will be the first to combine the following theoretical properties for a second order accurate numerical scheme for the ternary MMC system: (i) unique solvability and positivity-preserving property; (ii) energy stability; (iii) and optimal rate convergence. A few numerical results are also presented.

A positivity-preserving, second-order energy stable and convergent numerical scheme for a ternary system of macromolecular microsphere composite hydrogels

TL;DR

The authors develop a second-order, positivity-preserving, energy-stable finite difference scheme for a ternary MMC Cahn-Hilliard system governed by a Flory-Huggins–deGennes free energy. By combining a convex-concave energy decomposition with a BDF2 time discretization and Douglas-Dupont regularization, they ensure unique solvability, mass conservation, and unconditional energy dissipation, while maintaining pointwise positivity for and their sum. They establish a rigorous convergence theory, including higher-order consistency, rough and refined error estimates, yielding second-order convergence in and norms under a linear refinement between time step and mesh size, and even higher-order convergence under the enhanced consistency framework. Numerical experiments corroborate the theoretical properties, demonstrating mass conservation, energy decay, positivity, and the expected second-order accuracy, validating the scheme's reliability for simulating MMC hydrogels.

Abstract

A second order accurate numerical scheme is proposed and analyzed for the periodic three-component Macromolecular Microsphere Composite(MMC) hydrogels system, a ternary Cahn-Hilliard system with a Flory-Huggins-deGennes free energy potential. This numerical scheme with energy stability is based on the Backward Differentiation Formula(BDF) method in time derivation combining with Douglas-Dupont regularization term, combined the finite difference method in space. We provide a theoretical justification of positivity-preserving property for all the singular terms, i.e., not only the two phase variables are always between and , but also the sum of the two phase variables is between and , at a point-wise level. In addition, an optimal rate convergence analysis is provided in this paper, in which a higher order asymptotic expansion of the numerical solution, the rough error estimate and refined error estimate techniques have to be included to accomplish such an analysis. This paper will be the first to combine the following theoretical properties for a second order accurate numerical scheme for the ternary MMC system: (i) unique solvability and positivity-preserving property; (ii) energy stability; (iii) and optimal rate convergence. A few numerical results are also presented.
Paper Structure (18 sections, 4 theorems, 197 equations, 5 figures, 1 table)

This paper contains 18 sections, 4 theorems, 197 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. The fully discrete numerical scheme
  3. The finite difference spatial discretization
  4. A convex-concave decomposition of the discrete energy
  5. Unique solvability and positivity-preserving property
  6. The equivalent form of solving \ref{['Full-discrete-1']}-\ref{['Full-discrete-mu2']}
  7. Proof by contradiction
  8. The minimizer $(\phi_1^{\star},\phi_2^{\star})\in A_{h,\delta}$ could not occur at $\phi_1^{\star}=g(\delta)$.
  9. The minimizer $(\phi_1^{\star},\phi_2^{\star})\in A_{h,\delta}$ could not occur at $\phi_2^{\star}=g(\delta)$.
  10. The minimizer $(\phi_1^{\star},\phi_2^{\star})\in A_{h,\delta}$ could not occur at $\phi_1^{\star}+\phi_2^{\star}=1-\delta$.
  11. The minimizer $(\phi_1^{\star},\phi_2^{\star})\in A_{h,\delta}$ could not occur at $\phi_1^{\star}+\phi_2^{\star}=\delta$.
  12. Energy stability
  13. Optimal rate convergence analysis in $\ell^\infty (0,T; H_h^{-1}) \cap \ell^2 (0,T; H_h^1)$
  14. Higher order consistency analysis
  15. A rough error estimate
  16. ...and 3 more sections

Key Result

proposition 1

Suppose $(\phi_1,\phi_2) \in \vec{\mathcal{C}}_{\rm per}^{\mathcal{G}}$. The variational derivatives of $G_{h,c}$ and $G_{h,e}$ with respect to $\phi_1$ and $\phi_2$ are grid functions satisfying for $i = 1,2$.

Figures (5)

  • Figure 1: Example \ref{['example 2']}: Evolution of the energy over time. The time step size is taken as ${\Delta t} = 1.0 \times 10^{-4}$.
  • Figure 2: Example \ref{['example 2']}: The error developments of the total mass for $\phi_1$ and $\phi_2$, respectively.
  • Figure 3: Example \ref{['example 2']}: Evolution of three phase variables at $t = 10, 20, 30$ and $40$. The first line is for $\phi_1$, the second line is for $\phi_2$ and the last line is for $\phi_3$. The time step size is taken as ${\Delta t} = 1.0 \times 10^{-4}$.
  • Figure 4: Example \ref{['example 2']}: The time evolution of the maximum and minimum values for $\phi_1$ and $\phi_2$, respectively.
  • Figure 5: Example \ref{['example 2']}: The time evolutions of the maximum and minimum values for $\phi_1+\phi_2$, with ${\Delta t} = 1.0 \times 10^{-4}$.

Theorems & Definitions (6)

  • proposition 1
  • theorem 1
  • theorem 2
  • proof
  • proposition 2
  • proof