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

TL;DR

This paper rigorously proves the convergence of exponential time differencing schemes for the nonlocal Cahn-Hilliard equation, addressing challenges due to the absence of higher-order diffusion and validating results through numerical experiments.

Contribution

It introduces new error decomposition techniques and higher-order consistency analysis to establish convergence of ETD schemes for the NCH equation, which was previously unproven.

Findings

Established optimal convergence rates in relevant norms.

Validated schemes through numerical experiments.

Demonstrated effectiveness in long-term coarsening dynamics.

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.