In this paper, efficient alternating direction implicit (ADI) schemes are proposed to solve three-dimensional heat equations with irregular boundaries and interfaces. Starting from the well-known Douglas-Gunn ADI scheme, a modified ADI scheme is constructed to mitigate the issue of accuracy loss in solving problems with time-dependent boundary conditions. The unconditional stability of the new ADI scheme is also rigorously proven with the Fourier analysis. Then, by combining the ADI schemes with a 1D kernel-free boundary integral (KFBI) method, KFBI-ADI schemes are developed to solve the heat equation with irregular boundaries. In 1D sub-problems of the KFBI-ADI schemes, the KFBI discretization takes advantage of the Cartesian grid and preserves the structure of the coefficient matrix so that the fast Thomas algorithm can be applied to solve the linear system efficiently. Second-order accuracy and unconditional stability of the KFBI-ADI schemes are verified through several numerical tests for both the heat equation and a reaction-diffusion equation. For the Stefan problem, which is a free boundary problem of the heat equation, a level set method is incorporated into the ADI method to capture the time-dependent interface. Numerical examples for simulating 3D dendritic solidification phenomenons are also presented.

翻译：本文针对具有不规则边界与界面的三维热方程，提出了高效的交替方向隐式(ADI)格式。基于经典的Douglas-Gunn ADI格式，本文构建了一种改进的ADI格式，以缓解求解含时变边界条件问题时出现的精度损失问题。通过傅里叶分析严格证明了新ADI格式的无条件稳定性。进一步，将ADI格式与一维无核边界积分(KFBI)方法相结合，发展了KFBI-ADI格式用于求解不规则边界的热方程。在KFBI-ADI格式的一维子问题中，KFBI离散化充分利用笛卡尔网格并保持系数矩阵结构，从而可应用快速Thomas算法高效求解线性系统。通过热方程及反应扩散方程的多个数值试验，验证了KFBI-ADI格式的二阶精度与无条件稳定性。针对Stefan问题这一热方程的自由边界问题，将水平集方法融入ADI方法以捕捉时变界面。最后给出了三维枝晶凝固现象的数值模拟实例。