PDE-Based Multidimensional Extrapolation of Scalar Fields over Interfaces with Kinks and High Curvatures

TL;DR

This paper introduces a PDE-based method for multidimensional scalar field extrapolation across complex interfaces with kinks and high curvature, achieving high accuracy even at sharp geometric features.

Contribution

It proposes a novel extrapolation technique based on Cartesian derivatives, improving accuracy over existing methods near sharp interface features.

Findings

Achieves second- and third-order accuracy in $L^ abla$ norm with linear and quadratic extrapolations.

Demonstrates effectiveness in 2D and 3D examples, outperforming previous approaches.

Highlights importance of accurate extrapolation in solving PDEs on evolving domains.

Abstract

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 $L^\infty$ 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 [2]. The importance of accurate extrapolation near sharp geometric features is highlighted on an example of solving the diffusion equation on evolving domains.