AbstractThis work builds on a SIMPLE‐type code to produce two numerical codes of greatly improved speed and accuracy for solution of the Navier–Stokes equations. Both implicit and explicit codes employ an improved QUICK (quadratic upstream interpolation for convective kinematics) scheme to finite difference convective terms for non‐uniform grids. The PRIME (update pressure implicit, momentum explicit) algorithm is used as the computational procedure for the implicit code. Use of both the ICCG (incomplete Cholesky decomposition, conjugate gradient) method and the MG (multigrid) technique to enhance solution execution speed is illustrated. While the implicit code is first‐order in time, the explicit is second‐order accurate. Two‐ and three‐dimensional forced convection and sidewall‐heated natural convection flows in a cavity are chosen as test cases. Predictions with the new schemes show substantial computational savings and very good agreement when compared to previous simulations and ex