PDE-Based Multidimensional Extrapolation of Scalar Fields over Interfaces with Kinks and High Curvatures
Daniil Bochkov, Frédéric Gibou · SIAM Journal on Scientific Computing · 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.