Approximation error of the Lagrange reconstructing polynomial

TL;DR

This paper analyzes the approximation error of Lagrange reconstructing polynomials used in numerical methods for derivative approximation, providing explicit formulas and solutions for the deconvolution problem involved.

Contribution

It introduces an explicit solution to the deconvolution problem related to reconstruction pairs and derives the approximation error of Lagrange reconstructing polynomials on arbitrary stencils.

Findings

Explicit formulas for the deconvolution problem solution.

Derivation of the approximation error for Lagrange reconstructing polynomials.

Application of results to polynomial reconstruction on homogeneous grids.

Abstract

The reconstruction approach [Shu C.W.: {\em SIAM Rev.} {\bf 51} (2009) 82--126] for the numerical approximation of $f'(x)$ is based on the construction of a dual function $h(x)$ whose sliding averages over the interval $[x-\tfrac{1}{2}\Delta x,x+\tfrac{1}{2}\Delta x]$ are equal to $f(x)$ (assuming an homogeneous grid of cell-size $\Delta x$). We study the deconvolution problem [Harten A., Engquist B., Osher S., Chakravarthy S.R.: {\em J. Comp. Phys.} {\bf 71} (1987) 231--303] which relates the Taylor polynomials of $h(x)$ and $f(x)$, and obtain its explicit solution, by introducing rational numbers $\tau_n$ defined by a recurrence relation, or determined by their generating function, $g_\tau(x)$, related with the reconstruction pair of ${\rm e}^x$. We then apply these results to the specific case of Lagrange-interpolation-based polynomial reconstruction, and determine explicitly the approximation error of the Lagrange reconstructing polynomial (whose sliding averages are equal to the Lagrange interpolating polynomial) on an arbitrary stencil defined on a homogeneous grid.