Representation of the Lagrange reconstructing polynomial by combination of substencils

TL;DR

This paper analyzes the properties of Lagrange reconstructing polynomials on equidistant points, showing they can be uniquely expressed as combinations of substencils with rational weights, and proves the positivity of these weights near the midpoint.

Contribution

It introduces a unique representation of Lagrange reconstructing polynomials as a combination of substencils with explicit rational weights and proves their convexity near the midpoint.

Findings

Lagrange reconstructing polynomials have only real, distinct roots not at cell interfaces.

Existence of a unique combination of substencils for reconstructing polynomials.

Proof of convexity of weight-functions near the midpoint under certain conditions.

Abstract

The Lagrange reconstructing polynomial [Shu C.W.: {\em SIAM Rev.} {\bf 51} (2009) 82--126] of a function $f(x)$ on a given set of equidistant ($\Delta x=\const$) points $\bigl\{x_i+\ell\Delta x;\;\ell\in\{-M_-,...,+M_+\}\bigr\}$ is defined [Gerolymos G.A.: {\em J. Approx. Theory} {\bf 163} (2011) 267--305] as the polynomial whose sliding (with $x$) averages on $[x-\tfrac{1}{2}\Delta x,x+\tfrac{1}{2}\Delta x]$ are equal to the Lagrange interpolating polynomial of $f(x)$ on the same stencil. We first study the fundamental functions of Lagrange reconstruction, show that these polynomials have only real and distinct roots, which are never located at the cell-interfaces (half-points) $x_i+n\tfrac{1}{2}\Delta x$ ($n\in\mathbb{Z}$), and obtain several identities. Using these identities, by analogy to the recursive Neville-Aitken-like algorithm applied to the Lagrange interpolating polynomial, we show that there exists a unique representation of the Lagrange reconstructing polynomial on $\{i-M_-,...,i+M_+\}$ as a combination of the Lagrange reconstructing polynomials on the $K_\mathrm{s}+1\leq M:=M_-+M_+>1$ substencils $\{i-M_-+k_\mathrm{s},...,i+M_+-K_\mathrm{s}+k_\mathrm{s}\}$ ($k_\mathrm{s}\in\{0,...,K_\mathrm{s}\}$), with weights $\sigma_{R_1,M_-,M_+,K_\mathrm{s},k_\mathrm{s}}(\xi)$ which are rational functions of $\xi$ ($x=x_i+\xi\Delta x$) [Liu Y.Y., Shu C.W., Zhang M.P.: {\em Acta Math. Appl. Sinica} {\bf 25} (2009) 503--538], and give an analytical recursive expression of the weight-functions. We then use the analytical expression of the weight-functions $\sigma_{R_1,M_-,M_+,K_\mathrm{s},k_\mathrm{s}}(\xi)$ to obtain a formal proof of convexity (positivity of the weight-functions) in the neighborhood of $\xi=\tfrac{1}{2}$, under the condition that all of the substencils contain either point $i$ or point $i+1$ (or both).