Implicit treatment of advection terms and pressure correction
 

This question relates to a finite-difference code on a staggered grid similar to the one presened in the Kim and Moin JCP (1985) paper. In that paper the viscous terms are advanced implicitly using Adams Bashforth while the advection terms are advanced explicitly using Rung Kutta. Because of the implict treatment of the viscous diffusion terms the pressure is recovered as



If I want to treat implictly both the viscous diffusion and convection terms in one direction (in particular azimuthal terms in cylindrical coordinates) how do I recover the pressure?



Is there anything in the literature that addresses this?

Thank you in advance