G. A. Gerolymos

G. A. Gerolymos

In several applications, such as \tsc{weno} interpolation and reconstruction [Shu C.W.: SIAM Rev. 51 (2009) 82--126], we are interested in the analytical expression of the weight-functions which allow the representation of the approximating function on a given stencil (Chebyshev-system) as the weighted combination of the corresponding approximating functions on substencils (Chebyshev-subsystems). We show that the weight-functions in such representations [Mühlbach G.: Num. Math. 31 (1978) 97--110] can be generated by a general recurrence relation based on the existence of a 1-level subdivision rule. As an example of application we apply this recurrence to the computation of the weight-functions for Lagrange interpolation [Carlini E., Ferretti R., Russo G.: SIAM J. Sci. Comp. 27 (2005) 1071--1091] for a general subdivision of the stencil ${x_{i-M_-},...,x_{i+M_+}}$ of $M+1:=M_-+M_++1$ distinct ordered points into $K_\mathrm{s}+1\leq M:=M_-+M_+>1$ (Neville) substencils ${x_{i-M_-+k_\mathrm{s}},...,x_{i+M_+-K_\mathrm{s}+k_\mathrm{s}}}$ ($k_\mathrm{s}\in{0,...,K_\mathrm{s}}$) all containing the same number of $M-K_\mathrm{s}+1$ points but each shifted by 1 cell with respect to its neighbour, and give a general proof for the conditions of positivity of the weight-functions (implying convexity of the combination), extending previous results obtained for particular stencils and subdvisions [Liu Y.Y., Shu C.W., Zhang M.P.: Acta Math. Appl. Sinica 25 (2009) 503--538]. Finally, we apply the recurrence relation to the representation by combination of substencils of derivatives of arbitrary order of the Lagrange interpolating polynomial.

G. A. Gerolymos

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 $\bigl\{x_i+\ellΔx;\;\ell\in\{-M_-,...,+M_+\}\bigr\}$ is defined [Gerolymos G.A.: {\em J. Approx. Theory} {\bf 163} (2011) 267--305] as the polynomial whose sliding (with $x$) averages on $[x-\tfrac{1}{2}Δx,x+\tfrac{1}{2}Δx]$ 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) $x_i+n\tfrac{1}{2}Δx$ ($n\in\mathbb{Z}$), 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 $σ_{R_1,M_-,M_+,K_\mathrm{s},k_\mathrm{s}}(ξ)$ which are rational functions of $ξ$ ($x=x_i+ξΔx$) [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 $σ_{R_1,M_-,M_+,K_\mathrm{s},k_\mathrm{s}}(ξ)$ to obtain a formal proof of convexity (positivity of the weight-functions) in the neighborhood of $ξ=\tfrac{1}{2}$, under the condition that all of the substencils contain either point $i$ or point $i+1$ (or both).

G. A. Gerolymos

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 $[x-\tfrac{1}{2}Δx,x+\tfrac{1}{2}Δx]$ are equal to $f(x)$ (assuming an homogeneous grid of cell-size $Δx$). 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 $τ_n$ defined by a recurrence relation, or determined by their generating function, $g_τ(x)$, 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.