具有变迁移率和对数Flory-Huggins势的非局部Cahn-Hilliard系统的二阶保正格式

中文摘要 AI 辅助

本文分析了一种用于具有变迁移率和奇异Flory-Huggins对数势的非局部Cahn-Hilliard系统的二阶有限差分格式。时间离散采用修正的Crank-Nicolson格式，而迁移率被显式处理以保证椭圆性并降低计算成本。为了在离散层面严格保证保正性质，在数值格式中加入了一个非线性正则化项。严格分析表明，该格式在修正数值能量下具有唯一可解性和稳定性。与变迁移率相关的著名解析挑战需要谨慎处理。为克服这一挑战，基于两种非标准技术构建了一个收敛框架：（1）高阶渐近展开（在时间上延伸至三阶，在空间上延伸至四阶）以保持足够的精度阶数；（2）两阶段误差策略，其中粗略误差估计首先保证迁移率和非线性项的离散有界性，随后精细误差分析产生最优收敛速率。数值实验验证了理论结果并展示了所提格式的鲁棒性。

英文摘要

A second order finite difference scheme is analyzed for the nonlocal Cahn-Hilliard system with variable mobility and the singular Flory-Huggins logarithmic potential. The temporal discretization employs a modified Crank-Nicolson formulation, while the mobility is treated explicitly to guarantee ellipticity and reduce computational cost. To strictly enforce the positivity-preserving property at the discrete level, a nonlinear regularization term is added into the numerical scheme. Rigorous analysis shows that the scheme is uniquely solvable and stable under a modified numerical energy. The well-known analytical challenge associated with the variable mobility has to be handled carefully. To overcome this, a convergence framework is constructed based on two non-standard techniques: (1) a higher-order asymptotic expansion (extending up to the third order in time and fourth order in space) to retain a sufficiently order of accuracy; (2) a two-stage error strategy, wherein a rough error estimate first guarantees the discrete boundedness of the mobility and nonlinear terms, followed by a refined error analysis that yields the optimal convergence rate. Numerical experiments validate the theoretical results and demonstrate the robustness of the proposed scheme.

发表机构

• School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院）
• Laboratory of Mathematics and Complex Systems, Ministry of Education and School of Mathematical Sciences, Beijing Normal University（教育部数学与复杂系统重点实验室及北京师范大学数学科学学院）

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