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A refined convergence estimate for a fourth order finite difference numerical scheme to the Cahn-Hilliard equation

Jing Guo, Cheng Wang, Yue Yan, Xingye Yue

TL;DR

This work analyzes a second-order-in-time, fourth-order-in-space long-stencil finite difference scheme for the 3D Cahn-Hilliard equation, incorporating a modified BDF2 temporal discretization and Douglas-Dupont stabilization to guarantee energy stability. A key contribution is a refined error estimate in the discrete $H^{-1}$ norm, achieving $O(\Delta t^2 + h^4)$ accuracy with a convergence constant that depends polynomially on $\varepsilon^{-1}$ rather than exponentially on time, enabled by a spectrum-based analysis of the linearized CH operator and uniform-in-time Sobolev bounds. The authors develop a comprehensive suite of estimates for the fully discrete scheme, including higher-order Sobolev stability ($H_h^m$ with $m\ge2$), bounds for first- and second-order time differences, and precise handling of nonlinear terms via Fourier analysis and careful discrete-to-continuous comparisons. Numerical experiments in three dimensions validate the predicted convergence rates and demonstrate the expected energy dissipation, supporting the theoretical refinements and indicating practical impact for high-accuracy CH simulations.

Abstract

In this article we present a refined convergence analysis for a second order accurate in time, fourth order finite difference numerical scheme for the 3-D Cahn-Hilliard equation, with an improved convergence constant. A modified backward differentiation formula temporal discretization is applied, and a Douglas-Dupont artificial regularization is included to ensure the energy stability. In fact, a standard application of discrete Gronwall inequality leads to a convergence constant dependent on the interface width parameter in an exponential singular form. We aim to obtain an improved estimate, with such a singular dependence only in a polynomial order. A uniform in time functional bounds of the numerical solution, including the higher order Sobolev norms, as well as the associated bounds for the first and second order temporal difference stencil, have to be carefully established. Certain recursive analysis has to be applied in the analysis for the BDF-style temporal stencil. As a result, we are able to apply a spectrum estimate for the linearized Cahn-Hilliard operator, and this technique leads to the refined error estimate. A three-dimensional numerical example of accuracy check is presented as well.

A refined convergence estimate for a fourth order finite difference numerical scheme to the Cahn-Hilliard equation

TL;DR

This work analyzes a second-order-in-time, fourth-order-in-space long-stencil finite difference scheme for the 3D Cahn-Hilliard equation, incorporating a modified BDF2 temporal discretization and Douglas-Dupont stabilization to guarantee energy stability. A key contribution is a refined error estimate in the discrete norm, achieving accuracy with a convergence constant that depends polynomially on rather than exponentially on time, enabled by a spectrum-based analysis of the linearized CH operator and uniform-in-time Sobolev bounds. The authors develop a comprehensive suite of estimates for the fully discrete scheme, including higher-order Sobolev stability ( with ), bounds for first- and second-order time differences, and precise handling of nonlinear terms via Fourier analysis and careful discrete-to-continuous comparisons. Numerical experiments in three dimensions validate the predicted convergence rates and demonstrate the expected energy dissipation, supporting the theoretical refinements and indicating practical impact for high-accuracy CH simulations.

Abstract

In this article we present a refined convergence analysis for a second order accurate in time, fourth order finite difference numerical scheme for the 3-D Cahn-Hilliard equation, with an improved convergence constant. A modified backward differentiation formula temporal discretization is applied, and a Douglas-Dupont artificial regularization is included to ensure the energy stability. In fact, a standard application of discrete Gronwall inequality leads to a convergence constant dependent on the interface width parameter in an exponential singular form. We aim to obtain an improved estimate, with such a singular dependence only in a polynomial order. A uniform in time functional bounds of the numerical solution, including the higher order Sobolev norms, as well as the associated bounds for the first and second order temporal difference stencil, have to be carefully established. Certain recursive analysis has to be applied in the analysis for the BDF-style temporal stencil. As a result, we are able to apply a spectrum estimate for the linearized Cahn-Hilliard operator, and this technique leads to the refined error estimate. A three-dimensional numerical example of accuracy check is presented as well.
Paper Structure (21 sections, 15 theorems, 140 equations, 1 figure)

This paper contains 21 sections, 15 theorems, 140 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The numerical scheme and the main convergence result
  3. The spatial discretization and the related notations
  4. A few preliminary estimates
  5. The fully discrete numerical scheme and the energy stability
  6. Statement of the main theorem
  7. Higher order Sobolev estimates of the numerical scheme
  8. $\ell^\infty \left( 0, T ; H_h^m \right)$ ($m \ge 2$) bound of the scheme
  9. A uniform estimate for $\left\| \phi^{n+1} - \phi^n \right\|_{H_h^k}$
  10. A uniform estimate for $\left\| \phi^{n+1} - 2 \phi^n + \phi^{n-1} \right\|_{H_h^k}$
  11. Some established estimates for the exact solution and projection solution
  12. Error analysis with a refined convergence constant
  13. Consistency analysis and the numerical error evolutionary equation
  14. Review of the spectrum estimate for the linearized Cahn-Hilliard operator
  15. Proof of Theorem \ref{['thm:convergence']}
  16. ...and 6 more sections

Key Result

Lemma 2.1

cheng2019a We have

Figures (1)

  • Figure 1: The discrete $\ell^2$ and $\ell^\infty$ numerical errors versus spatial resolution $N$ for $N = 32:16:128$, and the time step size is set as ${\Delta t} = h^2$. The numerical results are obtained by the computation using the proposed numerical scheme \ref{['scheme-BDF-CH-1']}. The surface diffusion parameter is taken to be $\varepsilon=1$. The data lie roughly on curves $CN^{-4}$ for appropriate choices of $C$, confirming the full fourth order spatial accuracy and second order temporal accuracy.

Theorems & Definitions (23)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Corollary 2.7
  • Theorem 2.8
  • Theorem 3.1
  • proof
  • ...and 13 more