The flux reconstruction (FR) approach offers a flexible framework for describing a range of high-order numerical schemes; including nodal discontinuous Galerkin and spectral difference schemes. This is accomplished through the use of so-called correction functions. In this study we employ a weighted Sobolev norm to define a new extended family of FR correction functions, the stability of which is affirmed through Fourier analysis. Several of the schemes within this family are found to exhibit reduced dissipation and dispersion overshoot. Moreover, many of the new schemes possess higher CFL limits whilst maintaining the expected rate of convergence. Numerical experiments with homogeneous linear convection and Burgers’ turbulence are undertaken, and the results observed to be in agreement with the theoretical findings.

通量重构（FR）方法为描述一系列高阶数值方案提供了灵活的框架。包括节点间断Galerkin和频谱差异方案。这通过使用所谓的校正函数来实现。在这项研究中，我们采用加权Sobolev范数来定义FR校正函数的新扩展族，其稳定性通过傅立叶分析得到了证实。发现该系列中的几种方案显示出降低的耗散和色散过冲。此外，许多新方案在维持预期收敛速度的同时，具有更高的CFL限制。进行了具有均匀线性对流和伯格斯湍流的数值实验，并且观察到的结果与理论结果相符。