This paper proposes a Cartesian grid-based boundary integral method for efficiently and stably solving two representative moving interface problems, the Hele-Shaw flow and the Stefan problem. Elliptic and parabolic partial differential equations (PDEs) are reformulated into boundary integral equations and are then solved with the matrix-free generalized minimal residual (GMRES) method. The evaluation of boundary integrals is performed by solving equivalent and simple interface problems with finite difference methods, allowing the use of fast PDE solvers, such as fast Fourier transform (FFT) and geometric multigrid methods. The interface curve is evolved utilizing the $\theta-L$ variables instead of the more commonly used $x-y$ variables. This choice simplifies the preservation of mesh quality during the interface evolution. In addition, the $\theta-L$ approach enables the design of efficient and stable time-stepping schemes to remove the stiffness that arises from the curvature term. Ample numerical examples, including simulations of complex viscous fingering and dendritic solidification problems, are presented to showcase the capability of the proposed method to handle challenging moving interface problems.

翻译：本文提出一种基于笛卡尔网格的边界积分方法，用于高效稳定地求解两类代表性移动界面问题：Hele-Shaw流动和Stefan问题。将椭圆与抛物型偏微分方程重新表述为边界积分方程后，采用无矩阵广义最小残差法求解；边界积分评估则通过求解等价且简单的界面问题结合有限差分法实现，从而可运用快速傅里叶变换(Fast Fourier Transform, FFT)和几何多重网格法等快速偏微分方程求解器。界面曲线演化采用θ-L变量而非常用的x-y变量，这一选择简化了界面演化过程中网格质量的保持。此外，θ-L方法可设计高效稳定的时间步进格式以消除曲率项引发的刚性。文中通过包含复杂黏性指进和枝晶凝固问题模拟的大量数值算例，展示了所提方法处理挑战性移动界面问题的能力。