AI 中文总结
该研究对多维外推的静态偏微分方程方法进行再审视，提出变体方法，引入松弛迎风有限差分及边界重建技术，仅需边界相邻域内首网格层函数值等，提高计算效率与精度，经实验验证在多种域几何形状上性能增强。
AI 中文摘要
我们提出了一种基于静态偏微分方程的外推方法的变体，用于跨隐式定义为零水平集的界面扩展光滑场。我们的方法引入了松弛迎风有限差分的概念，作为快速扫描方法中的关键修改。与现有方法不同，我们仅需要在与边界相邻的域内的第一个网格层上的函数值及其法向导数，从而提高了计算效率和灵活性。为了进一步提高精度，我们引入了一种简单的边界重建技术，该技术显著降低了边界附近外推解中的数值误差。数值实验表明，我们的方法在一系列域几何形状上具有增强的性能，包括那些具有高曲率特征的几何形状。
英文摘要
We present a modified static PDE-based extrapolation method that builds on the approach of Aslam (2014 SIAM J. Sci. Comp) for extending smooth fields across interfaces implicitly defined as zero level sets. Our approach introduces the idea of relaxed upwinding finite differences as a key modification within the fast sweeping method. Unlike existing approaches, we require function values and their normal derivatives \emph{only} on the first grid layer inside the domain adjacent to the boundary, thereby improving computational efficiency and flexibility. To further improve accuracy, we introduce a simple boundary reconstruction technique that significantly reduces the numerical error in the extrapolated solution near the boundary. Numerical experiments indicate enhanced performance of our approach across a range of domain geometries, including those with high curvature features.
