An operator splitting scheme for the stream‐function formulation of unsteady Navier–Stokes equations

A fictitious time is introduced into the unsteady equation of the stream function rendering it into a higher‐order ultra‐parabolic equation. The convergence with respect to the fictitious time (we call the latter ‘internal iterations’) allows one to obtain fully implicit nonlinear scheme in full time steps for the physical‐time variable. For particular choice of the artificial time increment, the scheme in full time steps is of second‐order of approximation. For the solution of the internal iteration, a fractional‐step scheme is proposed based on the splitting of the combination of the Laplace, bi‐harmonic and advection operators. A judicious choice for the time staggering of the different parts of the nonlinear advective terms allows us to prove that the internal iterations are unconditionally stable and convergent. We assess the number of operations needed per time step and show computational effectiveness of the proposed scheme. We prove that when the internal iterations converge, the scheme is second‐order in physical time and space, nonlinear, implicit and absolutely stable. The performance of the scheme is demonstrated for the flow created by oscillatory motion of the lid of a square cavity. All theoretical findings are demonstrated practically. Copyright © 2006 John Wiley & Sons, Ltd.