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The Rhie--Chow [1] interpolation seems to be a standard tool in the Finite-Volume discretization of incompressible flows.

It is commonly defined on the discrete level [2].

In the lecture notes [3] -- Chapter 6, page 56 [2] -- it says that

We find that the Rhie-Chow interpolation is the same as adding a pressure term, which is proportional to a third [spatial] derivative. [...] In the continuity equation [...] the term is proportional to a fourth-order derivative term [...]

Is there any work on how this third or fourth order term actually looks like? Or is there an approximation of the NSE on the PDE level such that a, say, 1st order FVM discretization would directly lead to the Rhie--Chow interpolated scheme?

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