SIAM Journal on Scientific Computing, Volume 42, Issue 4, Page A2344-A2359, January 2020.
We present a PDE-based approach for the multidimensional extrapolation of smooth scalar quantities across interfaces with kinks and regions of high curvature. Unlike the commonly used method of [T. Aslam, J. Comput. Phys., 193 (2004), pp. 349--355], in which normal derivatives are extrapolated, the proposed approach is based on the extrapolation and weighting of Cartesian derivatives. As a result, second- and third-order accurate extensions in the $L^\infty$ norm are obtained with linear and quadratic extrapolations, respectively, even in the presence of sharp geometric features. The accuracy of the method is demonstrated on a number of examples in two and three spatial dimensions and compared to the approach of [T. Aslam, J. Comput. Phys., 193 (2004), pp. 349--355]. The importance of accurate extrapolation near sharp geometric features is highlighted on an example of solving the diffusion equation on evolving domains.

中文翻译：

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具有纠结和高曲率的接口上基于PDE的标量场的多维外推法

SIAM科学计算杂志，第42卷，第4期，第A2344-A2359页，2020年1月。
我们提出了一种基于PDE的方法，可对带有扭结和高曲率区域的界面上的平滑标量进行多维外推。与[T. Aslam，J.Comput。Phys。，193（2004），pp。349--355]，其中正态导数被外推，所提出的方法基于笛卡尔导数的外推和加权。结果，即使在存在尖锐的几何特征的情况下，也分别通过线性和二次外推获得了$ L ^ \ infty $范数中的二阶和三阶精确扩展。该方法的准确性在两个和三个空间维度上的多个示例中得到了证明，并与[T. Aslam，J.Comput。物理学报，193（2004），第349--355页。