In this paper, we develop an adaptive mesh refinement strategy of the Immersed Interface Method for flow problems with a moving interface. The work is built on the AMR method developed for two-dimensional elliptic interface problems in the paper [12] (CiCP, 12(2012), 515-527). The interface is captured by the zero level set of a Lipschitz continuous function φ(x, y, t). Our adaptive mesh refinement is built within a small band of |φ(x, y, t)| ≤ δ with finer Cartesian meshes. The AMR-IIM is validated for Stokes and Navier-Stokes equations with exact solutions, moving interfaces driven by the surface tension, and classical bubble deformation problems. A new simple area preserving strategy is also proposed in this paper for the level set method.

中文翻译：

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应用于流动问题的浸入式界面方法的自适应网格细化技术

在本文中，我们针对具有移动界面的流动问题开发了沉浸式界面方法的自适应网格细化策略。这项工作建立在论文 [12] (CiCP, 12(2012), 515-527) 中针对二维椭圆界面问题开发的 AMR 方法之上。该界面由 Lipschitz 连续函数 φ(x, y, t) 的零级集捕获。我们的自适应网格细化建立在 |φ(x, y, t)| 的小范围内。≤ δ 具有更精细的笛卡尔网格。AMR-IIM 已针对具有精确解的 Stokes 和 Navier-Stokes 方程、由表面张力驱动的移动界面和经典气泡变形问题进行了验证。本文还针对水平集方法提出了一种新的简单区域保留策略。