This work describes a detailed mathematical procedure in relation to a novel third‐order WENO scheme for the inviscid term of a system of nonlinear equations in the generalized grid system. The scheme developed minimizes the linear and nonlinear sources of dissipation error associated with the classical fifth‐order WENO scheme. The former is minimized by optimizing the resolving efficiency of the scheme whereas the latter is minimized by fixing the accuracy at the second‐order critical point via redefining the nonlinear weights. Moreover, the spectral property of second‐order viscous derivative, approximated by the single and double applications of the standard fourth‐order central finite difference scheme, is presented. The two‐dimensional Euler and Navier–Stokes equations in the generalized grids are mainly pursued. For the robustness in terms of capturing discontinuous and smooth structures, particularly two problems, which are difficult to handle in Cartesian grids, are chosen for discussion. The first one deals with a supersonic shock hitting the circular cylinder and generating all the possible flow inconsistencies. The other one deals with a subsonic flow over a circular cylinder at the incompressible limit. The numerical results are found to be in good agreement with the experimental data.

中文翻译：

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曲线网格的鲁棒低耗散方案

这项工作针对广义网格系统中非线性方程组的无形项，描述了与新颖的三阶WENO方案有关的详细数学过程。开发的方案最大程度地减少了与经典五阶WENO方案相关的线性和非线性耗散误差源。通过优化方案的求解效率将前者减至最小，而通过重新定义非线性权重将精度固定在二阶临界点，则将后者减至最小。此外，还给出了通过标准四阶中心有限差分方案的一次和两次应用近似得到的二阶粘性导数的光谱特性。广义网格中主要追求二维Euler和Navier-Stokes方程。为了在捕获不连续和光滑结构方面的鲁棒性，特别讨论了在笛卡尔网格中难以处理的两个问题。第一个方法是处理超声波冲击，撞击圆柱体并产生所有可能的流量不一致。另一个以不可压缩的极限处理圆柱体上的亚音速流。数值结果与实验数据吻合良好。另一个以不可压缩的极限处理圆柱体上的亚音速流。数值结果与实验数据吻合良好。另一个以不可压缩的极限处理圆柱体上的亚音速流。数值结果与实验数据吻合良好。