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Convergence analysis of exponential time differencing scheme for the nonlocal Cahn-Hilliard equation

Danni Zhang, Dongling Wang

TL;DR

This work addresses the convergence of fully discrete first- and second-order exponential time differencing schemes for the nonlocal Cahn–Hilliard equation under periodic boundary conditions, employing a Fourier spectral collocation discretization in space. By introducing two error-decomposition–based high-order consistency approaches and testing with $(-\Delta_N)^{-1}$ to compensate for the absent higher-order diffusion, the authors establish optimal convergence in the discrete $H_h^{-1}$ norm and in $\ell^2$ time, under mild mesh constraints. The results are complemented by rigorous proofs for both ETD1 and ETD2, including detailed higher-order consistency analyses and Gronwall-based error bounds, and are validated through comprehensive numerical experiments that exhibit convergence rates as well as long-time coarsening dynamics and energy evolution. This work provides reliable, high-accuracy time-stepping strategies for the NCH equation, enabling precise and efficient long-time simulations in applications across materials science and image processing.

Abstract

In this paper, we present a rigorous proof of the convergence of first order and second order exponential time differencing (ETD) schemes for solving the nonlocal Cahn-Hilliard (NCH) equation. The spatial discretization employs the Fourier spectral collocation method, while the time discretization is implemented using ETD-based multistep schemes. The absence of a higher-order diffusion term in the NCH equation poses a significant challenge to its convergence analysis. To tackle this, we introduce new error decomposition formulas and employ the higher-order consistency analysis. These techniques enable us to establish the $\ell^\infty$ bound of numerical solutions under some natural constraints. By treating the numerical solution as a perturbation of the exact solution, we derive optimal convergence rates in $\ell^\infty(0,T;H_h^{-1})\cap \ell^2(0,T; \ell^2)$. We conduct several numerical experiments to validate the accuracy and efficiency of the proposed schemes, including convergence tests and the observation of long-term coarsening dynamics.

Convergence analysis of exponential time differencing scheme for the nonlocal Cahn-Hilliard equation

TL;DR

This work addresses the convergence of fully discrete first- and second-order exponential time differencing schemes for the nonlocal Cahn–Hilliard equation under periodic boundary conditions, employing a Fourier spectral collocation discretization in space. By introducing two error-decomposition–based high-order consistency approaches and testing with to compensate for the absent higher-order diffusion, the authors establish optimal convergence in the discrete norm and in time, under mild mesh constraints. The results are complemented by rigorous proofs for both ETD1 and ETD2, including detailed higher-order consistency analyses and Gronwall-based error bounds, and are validated through comprehensive numerical experiments that exhibit convergence rates as well as long-time coarsening dynamics and energy evolution. This work provides reliable, high-accuracy time-stepping strategies for the NCH equation, enabling precise and efficient long-time simulations in applications across materials science and image processing.

Abstract

In this paper, we present a rigorous proof of the convergence of first order and second order exponential time differencing (ETD) schemes for solving the nonlocal Cahn-Hilliard (NCH) equation. The spatial discretization employs the Fourier spectral collocation method, while the time discretization is implemented using ETD-based multistep schemes. The absence of a higher-order diffusion term in the NCH equation poses a significant challenge to its convergence analysis. To tackle this, we introduce new error decomposition formulas and employ the higher-order consistency analysis. These techniques enable us to establish the bound of numerical solutions under some natural constraints. By treating the numerical solution as a perturbation of the exact solution, we derive optimal convergence rates in . We conduct several numerical experiments to validate the accuracy and efficiency of the proposed schemes, including convergence tests and the observation of long-term coarsening dynamics.
Paper Structure (14 sections, 4 theorems, 87 equations, 3 figures, 4 tables)

This paper contains 14 sections, 4 theorems, 87 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Fully discrete exponential time differencing schemes
  3. Convergence analysis
  4. Convergence analysis of ETD1
  5. Higher-order consistency analysis
  6. Convergence analysis in $\ell^\infty \left( 0,T;H_h^{-1} \right) \cap \ell^2 \left( 0,T; \ell^2 \right)$
  7. Convergence analysis of ETD2
  8. Higher-order consistency analysis
  9. Convergence analysis in $\ell^\infty(0,T;H_h^{-1})\cap \ell^2(0,T,\ell^2)$
  10. Numerical experiments
  11. Convergence tests
  12. Interfaces in the steady states
  13. Coarsening dynamics and energy evolution
  14. Conclusion remarks

Key Result

Lemma 2.1

(i) For $a>0$, the following inequalities hold: $0<(1+a)\phi_{-1}(a)<1,\, 1<(1+a)\phi_0(a)<\frac{3}{2}, \, \frac{1}{2}<(1+a)\phi_1(a)<1$ and $0<(1+a)[\phi_0(a)-\phi_1(a)]<1.$ (ii) If $0<s<\tau\leq 1$, then for any $a>0$, it holds $0<(1+a \tau)e^{-a (\tau-s)}<1.$

Figures (3)

  • Figure 1: Numerical simulation with $\delta=0.1$ for different values of $\varepsilon$.
  • Figure 3: Numerical results at $T=1,10,100,400,1200,2000$.
  • Figure 5: Numerical results at $T=0,50,200,500,700,1000$.

Theorems & Definitions (9)

  • Lemma 2.1: li2019
  • Lemma 2.2: Du1
  • Lemma 2.3: li1
  • Theorem 3.1
  • Example 4.1: 2D test
  • Example 4.2: 3D test
  • Example 4.3: 1D problem
  • Example 4.4: 2D problem
  • Example 4.5: 3D problem