Mesh-Robust Convergence of a Third-Order Variable-Step Deferred Correction Method for the Cahn–Hilliard Model

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发表日期：

2025年1月3日

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Nan Liu,
Jiahe Yue,
Hong-lin Liao

摘要

The variable-step deferred correction method constructs high-order schemes by modifying the low-order schemes, which not only relaxes the strict step-ratio constraints related to higher-order schemes, but also more accurately captures multi-scale evolution processes compared to lower-order schemes. It also provides internal error estimators to adjust the step sizes in adaptive time-stepping algorithms. Two third-order deferred correction methods based on the variable-step second-order BDF formula are analyzed for the Cahn–Hilliard model. The stability of the proposed variable-step deferred correction schemes is established under the step-ratio restriction of $0<r_k<4.864$ $0 < r_{k} < 4.864$ . By utilizing the discrete orthogonal convolution kernels and some discrete convolution embedding inequalities, the modified energy dissipation law and the $L^2$ $L^{2}$ norm error estimate at the discrete levels are established. Numerical experiments validate the effectiveness of our approaches.