A general recurrence relation for the weight-functions in M\"uhlbach-Neville-Aitken representions with application to WENO interpolation and differentiation

TL;DR

This paper introduces a general recurrence relation for weight-functions in M"uhlbach-Neville-Aitken representations, enabling analytical expressions for WENO interpolation and differentiation, with applications to Lagrange interpolation and positivity conditions.

Contribution

It presents a novel recurrence relation for weight-functions in Chebyshev-system representations, extending previous results and applying to WENO methods and derivatives of interpolating polynomials.

Findings

Derived a recurrence relation for weight-functions based on subdivision rules.

Extended positivity and convexity conditions for weight-functions.

Applied the recurrence to Lagrange interpolation and derivatives.

Abstract

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 [M\"uhlbach 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.