A non-separable progressive multivariate WENO-$2r$ point value

TL;DR

This paper introduces a new progressive multivariate WENO-$2r$ interpolation method that works with non-uniform data and multiple variables, providing high accuracy near discontinuities and explicit formulas for weights.

Contribution

It develops a general progressive WENO-$2r$ method for non-uniform multivariate data, including explicit weight formulas and theoretical accuracy proofs.

Findings

Achieves high-order accuracy in smooth regions

Maintains near-discontinuity accuracy with progressive order

Numerical experiments confirm theoretical results

Abstract

The weighted essentially non-oscillatory {technique} using a stencil of $2r$ points (WENO-$2r$) is an interpolatory method that consists in obtaining a higher approximation order from the non-linear combination of interpolants of $r+1$ nodes. The result is an interpolant of order $2r$ at the smooth parts and order $r+1$ when an isolated discontinuity falls at any grid interval of the large stencil except at the central one. Recently, a new WENO method based on Aitken-Neville's algorithm has been designed for interpolation of equally spaced data at the mid-points and presents progressive order of accuracy close to discontinuities. This paper is devoted to constructing a general progressive WENO method for non-necessarily uniformly spaced data and several variables interpolating in any point of the central interval. Also, we provide explicit formulas for linear and non-linear weights and prove the order obtained. Finally, some numerical experiments are presented to check the theoretical results.