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PDE-Based Multidimensional Extrapolation of Scalar Fields over Interfaces with Kinks and High Curvatures
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 [2] 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 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 [2]. The
importance of accurate extrapolation near sharp geometric features is
highlighted on an example of solving the diffusion equation on evolving
domains.Comment: 17 pages, 13 figures, submitted to SIAM Journal of Scientific
Computin