Convergence analysis of exponential time differencing scheme for the nonlocal Cahn-Hilliard equation
<|reference_start|>Convergence analysis of exponential time differencing scheme for the nonlocal Cahn-Hilliard equation: 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.<|reference_end|>
@article{zhang2024convergence,
    title={Convergence analysis of exponential time differencing scheme for the
  nonlocal Cahn-Hilliard equation},
    author={Danni Zhang and Dongling Wang},
    journal={arXiv preprint arXiv:2407.00572},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.00572},
    primaryClass={math.NA cs.NA}
}