  Two variants of the MCV3 scheme are presented based on a flux reconstruction
formulation. Different from the original multi-moment constrained finite volume
method of third order (MCV3), the multi-moment constraints are imposed at the
cell center on the point value, the first and second order derivatives. The
continuity of the flux function at cell interfaces are also used as the
constraints to ensure the numerical conservation. Compared to the original MCV3
scheme, both two variants have higher numerical accuracy and less restrictive
CFL condition for computational stability. Moreover, without the need to solve
derivative Riemann problem at cell boundaries, the new schemes benefit the
implementations in arbitrary quadrilateral in 2D and hexahedron in 3D.
