Representation of the Lagrange reconstructing polynomial by combination of substencils

Abstract

The Lagrange reconstructing polynomial [Shu C.W.: SIAM Rev. 51 (2009) 82-126] of a function f (x) on a given set of equidistant (∆x = const) points x i + ℓ∆x; ℓ ∈ {−M − , · · · , +M + } is defined as the polynomial whose sliding (with x) averages on [x − 1 2 ∆x, x + 1 2 ∆x] are equal to the Lagrange interpolating polynomial of f (x) on the same stencil [Gerolymos G.A.: J. Approx. Theory 163 (2011) 267-305]. 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 cellinterfaces (half-points) x i + n 1 2 ∆x (n ∈ Z), and obtain several identities. Using these identities, 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 Neville substencils [Carlini E., Ferretti R., Russo G.: SIAM J. Sci. Comp. 27 (2005) 1071-1091], with weights which are rational functions of ξ (x = x i + ξ∆x) [Liu Y.Y., Shu C.W., Zhang M.P.: Acta Math. Appl. Sinica 25 (2009) 503-538], and give an analytical recursive expression of the weight-functions.We show that all of the poles of the rational weight-functions are real, and that there can be no poles at half-points. We then use the analytical expression of the weight-functions, combined with the factorization of the fundamental functions of Lagrange reconstruction, to obtain a formal proof of convexity (positivity of the weight-functions) in the neighborhood of ξ = 1 2 , iff all of the substencils contain either point i or point i + 1 (or both).

CWENO: Uniformly accurate reconstructions for balance laws

“…WENO5 at cell centre). Results on the existence of d k (x) for general x have been proven for example in [5,12]. A procedure to circumvent the appearance of negative weights was proposed in [32].…”

“…It is easy to see that, for the sake of computing p(x), the zero-th order term in q(x) is not relevant. Thus the only divided differences that are needed are the ones listed in (12).…”

On the Accuracy of WENO and CWENO Reconstructions of Third Order on Nonuniform Meshes

“…One of the most successful high order reconstructions is the Weighted Essentially Non-Oscillatory (WENO), whose first efficient implementation was described in [12]. It is based on the observation that one may recover the value of high order interpolating polynomials on a centred stencil as convex combination of the values of lower order ones that interpolate the function only in a substencil (see [8] for a comprehensive development of this idea). In particular, let P j,λ (x) candidate polynomial in the j-th cell.…”

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

“…Liu et al [4] showed that the weight-functions, in the reconstruction case, are rational functions, expressed the weightfunctions for the (K s = M 2 )-level subdivision of the stencil {i − ⌊ M 2 ⌋, · · · , i + M − ⌊ M 2 ⌋}, in the range M ∈ {2, · · · , 11}, and computed the interval of convexity around x i+ 1 2 . In [9], we use the recurrence of Lemma 2.1 to study in detail the weight-functions for the Lagrange reconstructing polynomial [8], obtain explicit recursive expressions for the weightfunctions for an arbitrarily biased stencil on a homogeneous grid and an arbitrary subdivision level, and determine the interval of convexity in the neighbourhood of x i+ 1 2 . Liu et al [4], also study the representation of the first two derivatives (n ∈ {1, 2}) of the Lagrange interpolating polynomial for the particular homogeneous stencils and subdivisions studied in the reconstruction case (Remark 4.1).…”

“…One of the motivations that led to the formulation of Lemma 2.1 was the study of WENO reconstruction in view of the computation of f ′ (x), and this application is studied in [9] (results and relation to previous work are summarized in Remark 4.1). The expression of the (K s = 1)-level weight-functions for the Lagrange reconstructing polynomial is similar to (11), upon replacing the fundamental functions of Lagrange interpolation in (11) by the corresponding fundamental functions of Lagrange reconstruction [9, (32), Lemma 4.2, pp.…”