On the use of fast projection methods with unsteady velocity boundary conditions

This article aims at clarifying and amending a previously published result on the use of pseudo-pressure approximations for fast incompressible Navier-Stokes solvers with time-dependent boundary conditions. Fast projection methods were proposed to speed up high-order incompressible Navier-Stokes solvers by replacing pressure projections with simple approximations. A generalized approximation theory was proposed in [1], in which the time-dependent boundary conditions were ignored. In this comment, we investigate the effect of unsteady boundary conditions on the order of accuracy and demonstrate that the pressure approximations derived in [1] hold for all RK2 integrators, while RK3 approximations are only third order for certain sets of coefficients. We show that third order is lost for other cases and shed light on why order is broken. This work serves as a reference for the treatment of unsteady boundary conditions for fast-projection methods. Numerical validation is performed on a well established channel flow problem with a variety of unsteady inlet conditions for a wide range of explicit RK2 and RK3 integrators.