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投影导数计算公式构建。其高效性在于完全消除了对解变量或其他量的二阶导数进行计算和存储的需求，而这类二阶导数通常是将二阶单元中心非结构网格有限体积离散升级为三阶精度所必需的。此外，实现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，从而得到基于面心数值通量加上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基于单元中心源项评估值加上不涉及源项二阶导数的曲率修正项形式。该离散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，这带来了另一特性：无需为二次多项式计算和存储几何矩以保持单元平均值。本文通过简单但具有说明性的三维测试算例，验证并研究了所提出方法的三阶精度及改进的二阶精度。