The deferred correction method is a popular approach that enables to implement higher order (HO) convection schemes without losing out on stability issues such as diagonal dominance when solved implicitly. The approach evaluates HO terms explicitly by adding an implicit first order upwind (FOU) and subtracting the same but explicitly as shown in image 1 (Eq. 11.156).

Any HO flux interpolation can be represented by a linear combination of FOU scheme + HO scheme as shown in image 2. The first term in any HO scheme represents as FOU.

My question is can we not represent the HO scheme itself as implicit FOU + explicit gradient terms as shown in image 2 instead of representing it in the form given in Eq. (11.156)? Correct me if I am wrong, if you look at Eq. (11.156) the third term is evaluated as explicit FOU and does not this term cancel out from the first term of HO scheme that is also FOU? Any advantage (or consistency issues associated) with writing it in the form given by Eq. 11.156?

reference: The Finite Volume Method in Computational Fluid Dynamics

Book The Finite Volume Method in Computational Fluid Dynamics: An...