Representation of the Lagrange reconstructing polynomial by combination of substencils

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 ($Δx=\const$) points $\{x_i+\ell \Delta x;\ell \in \{-M_{-},...,+M_{+}\}\}$ is defined [Gerolymos G.A.: {\em J. Approx. Theory} {\bf 163} (2011) 267--305] as the polynomial whose sliding (with $x$) averages on $[x-\frac{1}{2}\Delta x,x+\frac{1}{2}\Delta x]$ 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) $x_i+n\frac{1}{2}\Delta x$ ($n\in \mathbb{Z}$), 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 $K_{\mathrm{s}}+1\le M:=M_{-}+M_{+}>1$ 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 $\xi =\frac{1}{2}$, under the condition that all of the substencils contain either point $i$ or point $i+1$ (or both).

final corrected version; in print J. Comp. Appl. Math

References in corpus (2)

• Approximation error of the Lagrange reconstructing polynomial
• A general recurrence relation for the weight-functions in Mühlbach-Neville-Aitken representions with application to WENO interpolation and differentiation

Cited by in corpus (3)

• On the accuracy of WENO and CWENO reconstructions of third order on nonuniform meshes
• CWENO: uniformly accurate reconstructions for balance laws
• A general recurrence relation for the weight-functions in Mühlbach-Neville-Aitken representions with application to WENO interpolation and differentiation