# BC's for intermediate velocities in Implicit Fractional Step Methods

Lately, I was reading some seminal papers on Fractional Step Algorithms and I found this one:

• Kim, D., Choi, H. A Second-Order Time-Accurate Finite Volume Method for Unsteady Incompressible Flow on Hybrid Unstructured Grids. Journal of Computational Physics, Volume 162, Issue 2, pp. 411-428 (2000)

And I am confused on as to how to use/define the boundary conditions for the auxiliary variables in the method proposed. Specifically, in this paper the use of a second auxiliary velocity is proposed, $\hat{u_i}$, and a certain boundary condition is proposed for it in equation (19). Anyway, the proposed integration method described by equations (12)-(15) seem not to require $\hat{u_i}$ on the boundaries; the rotational part of the momentum equation, equation (12), is solved for $\delta \hat{u}$ which, I think, it can be extrapolated from the cell-center to the faces (in specific, the boundary faces); and equation (13) only requires the surface integration of the left-hand side which only involves the pressure. My question is: Is the latter assertion (the extrapolation of $\delta \hat{u}$) correct? If not, How can I couple the boundary condition $\hat{u_i} = u_i^{n+1}+O(\Delta t^2)$ with equation (12) and equation (14)? How can be implemented?