  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 the 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.
