Mühlbach-Neville-Aitken表示中权重函数的一般递推关系及其在WENO插值与微分中的应用
A general recurrence relation for the weight-functions in Mühlbach-Neville-Aitken representions with application to WENO interpolation and differentiation
AI总结：
本文提出一种基于1级细分规则的通用递推关系来生成Mühlbach-Neville-Aitken表示中的权重函数，并应用于WENO插值和微分，同时证明了权重函数正性的条件。
摘要：
在若干应用中，例如WENO插值和重构[Shu C.W.: SIAM Rev. 51 (2009) 82--126]，我们关注权重函数的解析表达式，这些权重函数允许将给定模板（Chebyshev系统）上的逼近函数表示为子模板（Chebyshev子系统）上相应逼近函数的加权组合。我们证明，此类表示[Mühlbach G.: Num. Math. 31 (1978) 97--110]中的权重函数可以通过基于1级细分规则存在的通用递推关系生成。作为应用实例，我们将此递推关系应用于Lagrange插值[Carlini E., Ferretti R., Russo G.: SIAM J. Sci. Comp. 27 (2005) 1071--1091]中权重函数的计算，其中模板${x_{i-M_-},...,x_{i+M_+}}$（$M+1:=M_-+M_++1$个不同的有序点）被细分为$K_\mathrm{s}+1\leq M:=M_-+M_+>1$个（Neville）子模板${x_{i-M_-+k_\mathrm{s}},...,x_{i+M_+-K_\mathrm{s}+k_\mathrm{s}}}$（$k_\mathrm{s}\in{0,...,K_\mathrm{s}}$），每个子模板包含相同数量的$M-K_\mathrm{s}+1$个点，但相对于相邻子模板偏移1个单元，并给出权重函数正性条件（意味着组合的凸性）的一般证明，推广了先前针对特定模板和细分得到的结果[Liu Y.Y., Shu C.W., Zhang M.P.: Acta Math. Appl. Sinica 25 (2009) 503--538]。最后，我们将递推关系应用于通过子模板组合表示Lagrange插值多项式的任意阶导数。
英文摘要：
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.
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