Representation of the Lagrange reconstructing polynomial by combination of substencils

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• 作者：G.A. Gerolymos ; ^{georges.gerolymos@upmc.fr}
• 关键词：65D99 ; 65D05 ; 65D25 ; 65M06 ; 65M08
• 刊名：Journal of Computational and Applied Mathematics
• 出版年：2012
• 出版时间：May, 2012
• 年：2012
• 卷：236
• 期：11
• 页码：2763-2794
• 全文大小：1663 K

文摘

The Lagrange reconstructing polynomial [C.W. Shu, High-order WENO schemes for convection-dominated problems, SIAM Rev. 51 (1) (2009) 82-126] of a function on a given set of equidistant () points is defined as the polynomial whose sliding (with ) averages on are equal to the Lagrange interpolating polynomial of on the same stencil [G.A. Gerolymos, Approximation error of the Lagrange reconstructing polynomial, J. Approx. Theory 163 (2) (2011) 267-305. ]. We first study the fundamental functions of Lagrange reconstruction, then show that these polynomials have only real and distinct roots, which are never located at the cell-interfaces (half-points) (), and obtain several identities. Using these identities, we show that there exists a unique representation of the Lagrange reconstructing polynomial on as a combination of the Lagrange reconstructing polynomials on Neville substencils [E. Carlini, R. Ferretti, G. Russo, A WENO large time-step scheme for Hamilton-Jacobi equations, SIAM J. Sci. Comput. 27 (3) (2005) 1071-1091], with weights which are rational functions of () [Y.Y. Liu, C.W. Shu, M.P. Zhang, On the positivity of the linear weights in WENO approximations, Acta Math. Appl. Sin. 25 (3) (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 , iff all of the substencils contain either point or point (or both).