We present a high-order sharp treatment of immersed moving domain boundaries and material interfaces, and apply it to the advection-diffusion equation in two and three dimensions. The spatial discretization combines dimension-split finite difference schemes with an immersed boundary treatment based on a weighted least-squares reconstruction of the solution, providing stable discretizations with up to sixth order accuracy for diffusion terms and third order accuracy for advection terms. The temporal discretization relies on a novel strategy for maintaining high-order temporal accuracy in problems with moving boundaries that minimizes implementation complexity and allows arbitrary explicit or diagonally-implicit Runge-Kutta schemes. The approach is broadly compatible with popular PDE-specialized Runge-Kutta time integrators, including low-storage, strong stability preserving, and diagonally implicit schemes. Through numerical experiments we demonstrate that the full discretization maintains high-order spatial and temporal accuracy in the presence of complex 3D geometries and for a range of boundary conditions, including Dirichlet, Neumann, and flux conditions with large jumps in coefficients.

中文翻译：

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用于移动浸入域边界和材料界面的高阶有限差分方法

我们提出了浸没移动域边界和材料界面的高阶锐处理，并将其应用于二维和三维的平流扩散方程。空间离散化将维度分割有限差分方案与基于解的加权最小二乘重建的浸没边界处理相结合，为扩散项提供高达六阶精度的稳定离散化，为平流项提供高达三阶精度的稳定离散化。时间离散化依赖于一种新颖的策略，用于在移动边界的问题中保持高阶时间精度，从而最大限度地减少实现复杂性并允许任意显式或对角隐式龙格-库塔方案。该方法与流行的偏微分方程专用龙格库塔时间积分器广泛兼容，包括低存储、强稳定性保持和对角隐式方案。通过数值实验，我们证明，在存在复杂 3D 几何形状和一系列边界条件（包括狄利克雷、诺伊曼和系数跳跃较大的通量条件）的情况下，完全离散化可以保持高阶空间和时间精度。