2024/06/30 by Danni Zhang, Zhang, Danni, Dongling Wang +1 · 2 citations
Computer Science · Materials Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Solidification and crystal growth phenomena
paper · pdf · doi:10.48550/arxiv.2407.00572
openalex publication_date 2024/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ℓ^∞ 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 ℓ^∞(0,T;Hh-1)∩ ℓ2(0,T; ℓ2). 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.