In this paper, we propose an efficient quadratic interpolation formula utilizing solution gradients computed and stored at nodes and demonstrate its application to a third-order cell-centered finite-volume discretization on tetrahedral grids. The proposed quadratic formula is constructed based on an efficient formula of computing a projected derivative. It is efficient in that it completely eliminates the need to compute and store second derivatives of solution variables or any other quantities, which are typically required in upgrading a second-order cell-centered unstructured-grid finite-volume discretization to third-order accuracy. Moreover, a high-order flux quadrature formula, as required for third-order accuracy, can also be simplified by utilizing the efficient projected-derivative formula, resulting in a numerical flux at a face centroid plus a curvature correction not involving second derivatives of the flux. Similarly, a source term can be integrated over a cell to high-order in the form of a source term evaluated at the cell centroid plus a curvature correction, again, not requiring second derivatives of the source term. The discretization is defined as an approximation to an integral form of a conservation law but the numerical solution is defined as a point value at a cell center, leading to another feature that there is no need to compute and store geometric moments for a quadratic polynomial to preserve a cell average. Third-order accuracy and improved second-order accuracy are demonstrated and investigated for simple but illustrative test cases in three dimensions.

翻译：本文提出了一种利用节点处计算并存储的解梯度的高效二次插值公式，并展示了其在四面体网格上三阶单元中心有限体积离散中的应用。该二次公式基于一种高效的计算投影导数的公式构建。其高效性体现在完全消除了计算和存储解变量或其他量的二阶导数的需求，而这些在将二阶非结构网格有限体积离散升级到三阶精度时通常是必需的。此外，三阶精度所需的高阶通量求积公式也可通过利用该高效的投影导数公式进行简化，从而得到面心处的数值通量加上一个不涉及通量二阶导数的曲率修正项。类似地，源项在单元上的高阶积分可表示为单元形心处的源项值加上一个曲率修正项，同样不需要源项的二阶导数。离散形式定义为守恒律积分形式的近似，但数值解定义为单元中心的点值，这带来了另一个特点：无需计算和存储用于保持单元平均值的二次多项式的几何矩。通过简单但具有代表性的三维测试算例，展示并研究了该方法的三阶精度和改善的二阶精度。