This paper extends the gradient-based reconstruction approach of Chamarthi \cite{chamarthi2023gradient} to genuine high-order accuracy for inviscid test cases involving smooth flows. A seventh-order accurate scheme is derived using the same stencil as of the explicit fourth-order scheme proposed in Ref. \cite{chamarthi2023gradient}, which also has low dissipation properties. The proposed method is seventh-order accurate under the assumption that the variables at the \textit{cell centres are point values}. A problem-independent discontinuity detector is used to obtain high-order accuracy. Accordingly, primitive or conservative variable reconstruction is performed around regions of discontinuities, whereas smooth solution regions apply flux reconstruction. The proposed approach can still share the derivatives between the inviscid and viscous fluxes, which is the main idea behind the gradient-based reconstruction. Several standard benchmark test cases are presented. The proposed method is more efficient than the seventh-order weighted compact nonlinear scheme (WCNS) for the test cases considered in this paper.

翻译：本文将Chamarthi \cite{chamarthi2023gradient}提出的基于梯度的重构方法扩展至真正的高阶精度，适用于含光滑流动的无粘算例。采用与文献\cite{chamarthi2023gradient}中显式四阶格式相同的模板，推导出具有低耗散特性的七阶精度格式。在假设变量为\textit{单元中心点值}的前提下，该方法达到七阶精度。采用与问题无关的不连续性检测器实现高阶精度，即在不连续区域附近进行原始变量或守恒变量重构，而在光滑解区域应用通量重构。该方法仍可在无粘通量与粘性通量之间共享导数，这正是基于梯度重构的核心思想。通过多个标准基准算例验证表明，对于本文所考虑的算例，所提方法较七阶加权紧致非线性格式(WCNS)具有更高效率。