Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Bound Preserving Energy Stable Scheme for a Nonlocal Cahn-Hilliard Equation

Rainey Lyons, Grigor Nika, Adrian Muntean

TL;DR

This work develops a finite-volume, bound-preserving, energy-stable numerical scheme for a nonlocal Cahn–Hilliard equation formulated as a gradient flow with degenerate mobility $M(\rho)=\beta(1+\rho)(1-\rho)$. The method combines convex-splitting ideas for the local energy with a nonlocal convolution implemented via a circular convolution, ensuring mass conservation, $|\rho|\le 1$, and a discrete energy dissipation mechanism through a pseudo-energy $\hat{\mathcal{E}}_h$. The authors prove the key properties and validate them numerically for Flory–Huggins and Ginzburg–Landau potentials, observing robust phase separation dynamics and consistent energy decay. The framework provides a structure-preserving tool for simulating morphology formation in multi-component systems and sets the stage for extensions to more complex models derived from Kawasaki dynamics.

Abstract

We present a finite-volume based numerical scheme for a nonlocal Cahn-Hilliard equation which combines ideas from recent numerical schemes for gradient flow equations and nonlocal Cahn-Hilliard equations. The equation of interest is a special case of a previously derived and studied system of equations which describes phase separation in ternary mixtures. We prove the scheme is both energy stable and respects the analytical bounds of the solution. Furthermore, we present numerical demonstrations of the theoretical results using both the Flory-Huggins (FH) and Ginzburg-Landau (GL) free-energy potentials.

A Bound Preserving Energy Stable Scheme for a Nonlocal Cahn-Hilliard Equation

TL;DR

This work develops a finite-volume, bound-preserving, energy-stable numerical scheme for a nonlocal Cahn–Hilliard equation formulated as a gradient flow with degenerate mobility . The method combines convex-splitting ideas for the local energy with a nonlocal convolution implemented via a circular convolution, ensuring mass conservation, , and a discrete energy dissipation mechanism through a pseudo-energy . The authors prove the key properties and validate them numerically for Flory–Huggins and Ginzburg–Landau potentials, observing robust phase separation dynamics and consistent energy decay. The framework provides a structure-preserving tool for simulating morphology formation in multi-component systems and sets the stage for extensions to more complex models derived from Kawasaki dynamics.

Abstract

We present a finite-volume based numerical scheme for a nonlocal Cahn-Hilliard equation which combines ideas from recent numerical schemes for gradient flow equations and nonlocal Cahn-Hilliard equations. The equation of interest is a special case of a previously derived and studied system of equations which describes phase separation in ternary mixtures. We prove the scheme is both energy stable and respects the analytical bounds of the solution. Furthermore, we present numerical demonstrations of the theoretical results using both the Flory-Huggins (FH) and Ginzburg-Landau (GL) free-energy potentials.
Paper Structure (9 sections, 3 theorems, 39 equations, 4 figures)

This paper contains 9 sections, 3 theorems, 39 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Model and assumptions
  3. Numerical Scheme
  4. Discretization
  5. Circular Convolution
  6. Bound Preserving Scheme
  7. Main results
  8. Simulation
  9. Conclusion

Key Result

Theorem 3.1

For any $\Delta t, h > 0$, scheme Eq:CL_Scheme has the following properties:

Figures (4)

  • Figure 1: Maximum value of $|m|$ over time when solving \ref{['Eq:MarraPDE']} with a finite-volume method without using flux limiters. Notice here that the analytical bound $|m| \leq 1$ is not maintained.
  • Figure 2: Simulation of phase separation with the Flory--Huggins potential (top) and Ginzburg--Landau potential (bottom) using scheme \ref{['Eq:CL_Scheme']} for $N = 2^7$, $\beta = 5$, and with a time step $\Delta t = 10^{-2}$. Here, phases near 1 are colored blue, near -1 yellow, and near 0 red.
  • Figure 3: Left: Evolution of the free energy \ref{['Eq:Free_Energy']} over time for both the Flory--Huggins (FH) and Ginzburg--Landau (GL) potentials. Right: Maximum value of $|\rho|$ over time.
  • Figure 4: Energy dissipation rate for the Flory--Huggins (FH) and Ginzburg--Landau (GL) potentials.

Theorems & Definitions (7)

  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.1
  • Theorem 3.3
  • proof