  This paper introduces a novel approach to compute the numerical fluxes at the
cell boundaries in the finite volume approach. Explicit gradients used in
deriving the reconstruction polynomials are replaced by high-order gradients
computed by compact finite differences, referred to as implicit gradients in
this paper. The new finite volume scheme has superior dispersion and
dissipation properties in comparison to the compact reconstruction approach.
These implicit gradients are re-used in viscous flux computation and
post-processing, which further improves efficiency. A problem-independent shock
capturing approach via Boundary Variation Diminishing (BVD) algorithm is used
to suppress oscillations for the simulation of flows with shocks and material
interfaces. Several numerical test cases are carried out to verify the proposed
finite volume method's capability using the implicit gradient method for single
and multicomponent flows. Significant improvements are observed by computing
the gradients implicitly for the viscous flows.
