A Comparative Study of Time-Stepping Techniques for the Incompressible Navier-Stokes Equations: From Fully Implicit Non-Linear Schemes to Semi-Implicit Projection Methods

We present a numerical comparison of some time-stepping schemes for the discretization and solution of the non-stationary incompressible Navier– Stokes equations. The spatial discretization is by non-conforming quadrilateral finite elements which satisfy the LBB condition. The major focus is on the differences in accuracy and efficiency between the backward Euler, Crank–Nicolson and fractional-step Θ schemes used in discretizing the momentum equations. Further, the differences between fully coupled solvers and operator-splitting techniques (projection methods) and the influence of the treatment of the nonlinear advection term are considered. The combination of both discrete projection schemesand non-conforming finite elementsallows the comparison of schemes which are representative for many methods used in practice. On Cartesian grids this approach encompasses some well-known staggered grid finite difference discretizations too. The results which are obtained for several typical flow problems are thought to be representative and should be helpful for a fair rating of solution schemes, particularly in long-time simulations.