  This study introduces an order-lifted inversion/retrieval method for
implementing high-order schemes within the framework of an
unstructured-mesh-based finite-volume method. This method defines a special
representation called the data order-lifted inversion of neighbor cells
(DOLINC) differential, which transforms the degrees of freedom of wide
templates into differentials of various orders stored in local grid cells.
Furthermore, to retrieve the original far-field information without bias during
the reconstruction/interpolation of face values, the corresponding accurate
inversion formulas are derived based on the defined DOLINC differentials. The
order-lifted inversion method can be applied to multi-dimensional
polyhedral-mesh solvers by considering the influence of grid non-uniformity on
high-order schemes. It seamlessly accommodates multi-process parallel computing
for high-order methods without requiring special consideration for the boundary
interface. This method not only enhances the numerical accuracy of second-order
finite-volume methods, but also demonstrates a significant computational-speed
advantage over similar methods. A series of benchmark cases, including the
linear advection, Burgers, and Euler equations, are comprehensively validated
to assess the practical performance of the method. The results indicate that
the unstructured-mesh high-order schemes implemented based on this method
achieve theoretical accuracy in practical computations and substantially reduce
computational costs compared with methods that increase grid resolution.
