拉格朗日重构多项式通过子模板组合的表示
Representation of the Lagrange reconstructing polynomial by combination of substencils
AI总结：
本文研究了拉格朗日重构多项式的基本函数，证明了其根为实根且互异，并给出了该多项式可唯一表示为子模板上拉格朗日重构多项式组合的解析递归表达式，同时证明了在特定条件下权重函数在ξ=1/2邻域内的凸性。
摘要：
函数$f(x)$在给定等距点集$\bigl\{x_i+\ell\Delta x;\;\ell\in\{-M_-,...,+M_+\}\bigr\}$上的拉格朗日重构多项式[Shu C.W.: {\em SIAM Rev.} {\bf 51} (2009) 82--126]定义为这样一个多项式，其在$[x-\tfrac{1}{2}\Delta x,x+\tfrac{1}{2}\Delta x]$上的滑动（随$x$）平均值等于同一模板上$f(x)$的拉格朗日插值多项式[Gerolymos G.A.: {\em J. Approx. Theory} {\bf 163} (2011) 267--305]。我们首先研究了拉格朗日重构的基本函数，证明了这些多项式只有实根且互异，这些根永远不会位于单元界面（半整数点）$x_i+n\tfrac{1}{2}\Delta x$（$n\in\mathbb{Z}$）处，并得到了若干恒等式。利用这些恒等式，类比于应用于拉格朗日插值多项式的递归Neville-Aitken算法，我们证明了在$\{i-M_-,...,i+M_+\}$上的拉格朗日重构多项式可以唯一表示为$K_\mathrm{s}+1\leq M:=M_-+M_+>1$个子模板$\{i-M_-+k_\mathrm{s},...,i+M_+-K_\mathrm{s}+k_\mathrm{s}\}$（$k_\mathrm{s}\in\{0,...,K_\mathrm{s}\}$）上的拉格朗日重构多项式的组合，权重$\sigma_{R_1,M_-,M_+,K_\mathrm{s},k_\mathrm{s}}(\xi)$是$\xi$（$x=x_i+\xi\Delta x$）的有理函数[Liu Y.Y., Shu C.W., Zhang M.P.: {\em Acta Math. Appl. Sinica} {\bf 25} (2009) 503--538]，并给出了权重函数的解析递归表达式。然后，我们利用权重函数$\sigma_{R_1,M_-,M_+,K_\mathrm{s},k_\mathrm{s}}(\xi)$的解析表达式，在子模板均包含点$i$或点$i+1$（或两者）的条件下，形式化证明了在$\xi=\tfrac{1}{2}$邻域内的凸性（权重函数的正性）。
英文摘要：
The Lagrange reconstructing polynomial [Shu C.W.: {\em SIAM Rev.} {\bf 51} (2009) 82--126] of a function $f(x)$ on a given set of equidistant ($Δx=\const$) points $\bigl\{x_i+\ellΔx;\;\ell\in\{-M_-,...,+M_+\}\bigr\}$ is defined [Gerolymos G.A.: {\em J. Approx. Theory} {\bf 163} (2011) 267--305] as the polynomial whose sliding (with $x$) averages on $[x-\tfrac{1}{2}Δx,x+\tfrac{1}{2}Δx]$ are equal to the Lagrange interpolating polynomial of $f(x)$ on the same stencil. 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 cell-interfaces (half-points) $x_i+n\tfrac{1}{2}Δx$ ($n\in\mathbb{Z}$), and obtain several identities. Using these identities, by analogy to the recursive Neville-Aitken-like algorithm applied to the Lagrange interpolating polynomial, 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 the $K_\mathrm{s}+1\leq M:=M_-+M_+>1$ substencils $\{i-M_-+k_\mathrm{s},...,i+M_+-K_\mathrm{s}+k_\mathrm{s}\}$ ($k_\mathrm{s}\in\{0,...,K_\mathrm{s}\}$), with weights $σ_{R_1,M_-,M_+,K_\mathrm{s},k_\mathrm{s}}(ξ)$ which are rational functions of $ξ$ ($x=x_i+ξΔx$) [Liu Y.Y., Shu C.W., Zhang M.P.: {\em Acta Math. Appl. Sinica} {\bf 25} (2009) 503--538], and give an analytical recursive expression of the weight-functions. We then use the analytical expression of the weight-functions $σ_{R_1,M_-,M_+,K_\mathrm{s},k_\mathrm{s}}(ξ)$ to obtain a formal proof of convexity (positivity of the weight-functions) in the neighborhood of $ξ=\tfrac{1}{2}$, under the condition that all of the substencils contain either point $i$ or point $i+1$ (or both).
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