不规则域上Allen-Cahn与Cahn-Hilliard方程的无核边界积分方法

中文摘要 AI 辅助

针对二维和三维不规则域上具有齐次无通量边界条件的Allen-Cahn和Cahn-Hilliard方程，提出了一种统一的无核边界积分（KFBI）框架。采用稳定的一阶隐式-显式（IMEX）离散化后，Allen-Cahn更新简化为Neumann修正Helmholtz问题。通过辅助变量重构，Cahn-Hilliard更新被简化为同类型的子问题，无需评估非线性源的Laplacian。Cahn-Hilliard更新需要对实位移进行两次顺序求解，而复共轭位移允许从单次复求解中重构实值解。所有子问题均由同一KFBI求解器处理，该求解器通过等效界面问题间接评估势能。为修正Cahn-Hilliard更新中的离散质量缺陷，构建了针对不规则边界KFBI处理的几何加权离散$L^2$质量投影。外部重叠权重被重新分配到物理节点，将辅助KFBI扩展值排除在质量计算之外。数值实验证明了二阶空间收敛性、成对时间误差指标的一阶收敛性以及规定离散质量的保持。所报告的源自由模拟展示了能量衰减，累积投影引起的能量扰动相对于观测到的耗散保持较小。

英文摘要

A unified kernel-free boundary integral (KFBI) framework is proposed for the Allen-Cahn and Cahn Hilliard equations with homogeneous no-flux boundary conditions on two- and three-dimensional irregular domains. With a stabilized first-order implici-explicit (IMEX) discretization, the Allen-Cahn update reduces to a Neumann modified Helmholtz problem. An auxiliary-variable reformulation reduces the Cahn-Hilliard update to subproblems of the same type without evaluating the Laplacian of the nonlinear source. The Cahn-Hilliard update requires two sequential solves for real shifts, while complex-conjugate shifts allow the real-valued solution to be reconstructed from a single complex solve. All subproblems are handled by the same KFBI solver that indirectly evaluates potentials through equivalent interface problems. To correct discrete mass defects in the Cahn-Hilliard update, a geometry-weighted discrete $Lš$ mass projection is constructed tailored to the KFBI treatment of irregular boundaries. Exterior overlap weights are redistributed to physical nodes, excluding auxiliary KFBI extension values from the mass calculation. Numerical experiments demonstrate second-order spatial convergence, first order convergence of paired temporal-error indicators, and preservation of the prescribed discrete mass. The reported source-free simulations exhibit energy decay, with cumulative projection-induced energy perturbations remaining small relative to the observed dissipation.

发表机构

• School of Mathematical Sciences and Institute of Natural Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院和自然科学研究院）
• School of Mathematical Sciences, MOE-LSC and Institute of Natural Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院、教育部拉格朗日计算科学中心和自然科学研究院）

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