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

TL;DR

This paper tackles Gibbs-type oscillations in high-order WENO interpolation on non-uniform grids and in multiple dimensions by introducing a progressive, Aitken-Neville-based WENO framework. The authors derive a univariate progressive $WENO$-$2r$ algorithm for non-uniform data, establish explicit linear and nonlinear weights, and prove that the method achieves $O(h^{2r})$ accuracy in smooth regions and controlled $O(h^{r+l_0})$ accuracy near isolated discontinuities. They extend the construction to multivariate settings, first detailing Cartesian-grid tensor-product Lagrange interpolation, then developing a new progressive bivariate and finally a general progressive multivariate WENO method with explicit weight formulas and a generalized smoothness-indicator design. Numerical experiments in 1D and 2D, on both uniform and non-uniform grids, confirm the theoretical orders and demonstrate robust Gibbs-phenomenon avoidance, highlighting improved local accuracy near discontinuities compared to classical WENO. The work provides a practical, dimension-agnostic interpolation framework with explicit weight structures, paving the way for efficient PDE applications on non-uniform meshes.

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.

Figures (8)

• Figure 1: Diagram showing the structure of the optimal weights needed to obtain optimal order of accuracy, ARSY20.
• Figure 2: Stencils used to get $p^{3}_{\mathbf{j}_1}(\mathbf{x}^*), \, \mathbf{j}_1\in \{0,1,2\}^2$. They are used in classical bivariate WENO arandigamuletrenau
• Figure 3: Stencils used to get $p^4_{\mathbf{j}_0}(\mathbf{x}^*)$, $\mathbf{j}_0\in\{0,1\}^2$.
• Figure 4: Contribution of each stencil of $p^{3}_{\mathbf{j}_1}(\mathbf{x}^*)$, $\mathbf{j}_1\in \mathbf{j}_1+\{0,1\}^2$ to the approximation of $p^4_{\mathbf{j}_0}(\mathbf{x}^*)$, $\mathbf{j}_0\in\{0,1\}^2$.
• Figure 5: Interpolation using the new WENO-6 algorithm. Dashed line: interpolation of the function \ref{['experimento1']} (a) and of the function \ref{['experimento2']} (b). Circles: Data points.

Theorems & Definitions (11)

• Lemma 2.1
• Corollary 2.2
• Definition 1
• Proposition 2.3
• Theorem 2.4
• Theorem 3.1
• Lemma 4.1
• Corollary 4.2
• Theorem 5.1
• Theorem 6.1