A general recurrence relation for the weight-functions in Mühlbach–Neville–Aitken representations with application to WENO interpolation and differentiation

Abstract

In several applications, such as WENO interpolation and reconstruction [Shu C.W.: SIAM Rev. 51 (2009) , 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 {xcontaining the same number of M − K 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.

Representation of the Lagrange reconstructing polynomial by combination of substencils

“…For both cases it is shown [6,11] that ∀ξ ∈ [−1, 1] the linear weights are positive, and as a consequence the above combination of substencils is convex ∀ξ ∈ [−1, 1]. In a recent work [12] we extended these results for the general K s -level subdivision of an arbitrary stencil X i−M − ,i+M + : [13] studies Chebyshev-systems satisfying interpolatory conditions. In the reconstructing polynomial case (5a), the usual linear system approach [10, (13) [12, (4e), Lemma 2.1], only requires that the (K s = 1)-level subdivision can be defined.…”

“…The analytical results obtained in the present work, in particular the recursive analytical expression of the weightfunctions σ R 1 ,M − ,M + ,K s ,k s (ξ) (Proposition 4.5) and the factorization of the fundamental functions of Lagrange reconstruction α R 1 ,M − ,M + ,ℓ (ξ) (Proposition 3.7), can be used to study convexity intervals for arbitrary values of [M ± , K s ], as was recently done for the Lagrange interpolating polynomial [12,Proposition 3.2]. In [7, Result 6.1, p. 300] we had conjectured that for any choice of [M ± , K s ] for which all of the substencils S i,M − −k s ,M + −K s +k s (k s ∈ {0, · · · , K s }) contain either point i or point i + 1 (or both), convexity was observed at ξ = 1 2 .…”

“…3 Liu et al [11] have concentrated on the usual WENO substencils. 4 In the reconstruction case, it was shown by construction [11] that the optimal weight-functions are not polynomials, as in the interpolation case [6,11,12], but, instead, rational functions of ξ (x = x i + ξ∆x), implying that at the poles of these rational functions the weight-functions cannot be defined. For upwind-biased schemes [2,9,10] the big stencil {i − (r − 1), · · · , i + (r − 1)} (r ∈ N ≥2 ) which is centered around the point i, and upwind-biased with respect to the cell-face i + 1 2 , is subdivided [11, Tab.…”

“…These weights were further analyzed to determine the regions of convexity of the representation (positivity of the weight-functions). These important results [11] include explicit expressions of the weight-functions for the particular stencils which were studied, but a general analytical expression of the optimal weight-functions for the representation of the Lagrange reconstructing polynomial by combination of substencils is not yet available, contrary to the interpolating polynomial case [6,11,12]. The work of Liu et al [11] is based on the reconstruction via primitive approach [16, pp.…”

Further analysis of the budgets of the dissipation tensor ${\varepsilon }_{{ij}}$ in turbulent plane channel flow

“…Correlations in (1b) were computed using order-4 inhomogeneous-grid interpolating polynomials (Gerolymos, 2012) and sampled at every iteration (∆t + s = ∆t + 0.0059) for an observation interval t + OBS 1113. Because of the relatively short observation interval, the pressure term Π εij (1b) which contains the highly intermittent pressure-Hessian (Vreman and Kuerten, 2014b, Fig.…”