Nodal point sets, and associated collocation projections, play an important role in a range of high-order methods, including Flux Reconstruction (FR) schemes. Historically, efforts have focused on identifying nodal point sets that aim to minimise the $L^{\infty }$ error of an associated interpolating polynomial. The present work combines a comprehensive review of known approximation theory results, with new results, and numerical experiments, to motivate that in fact point sets for FR should aim to minimise the $L^2$ error of an associated interpolating polynomial. New results include identification of a nodal point set that minimises the $L^2$ norm of an interpolating polynomial, and a proof of the equivalence between such an interpolating polynomial and an $L^2$ approximating polynomial with coefficients obtained using a Gauss–Legendre quadrature rule. Numerical experiments confirm that FR errors can be reduced by an order-of-magnitude by switching from popular point sets such as Chebyshev, Chebyshev–Lobatto and Legendre–Lobatto to Legendre point sets.

节点集和相关的搭配预测在一系列高阶方法（包括磁通重构（FR）方案）中起着重要作用。从历史上看，工作重点一直放在确定节点集上，以最大程度地减少${\mathrm{大号}}^{\infty }$相关插值多项式的误差。目前的工作结合了对已知近似理论结果的全面综述，新的结果和数值实验，以激发事实上FR的点集应旨在最大程度地减少${\mathrm{大号}}^2$相关插值多项式的误差。新的结果包括识别节点集，以最大程度地减少${\mathrm{大号}}^2$ 插值多项式的范数，以及该插值多项式与 ${\mathrm{大号}}^2$近似多项式，其系数使用高斯-勒格德勒正交规则获得。数值实验证实，通过从常用的点集（例如Chebyshev，Chebyshev-Lobatto和Legendre-Lobatto）切换到Legendre点集，可以将FR误差减少一个数量级。