Numerical Study of a Modified Time-Stepping θ-Scheme for Incompressible Flow Simulations

In Turek (1996). Int. J. Numer. Meth. Fluids 22, 987–1011], we had performed numerical comparisons for different time stepping schemes for the incompressible Navier–Stokes equations. In this paper, we present the numerical analysis in the context of the Navier–Stokes equations for a modified time-stepping θ-scheme which has been recently proposed by Glowinski Glowinski (2003). In: Ciarlet, P. G., and Lions, J. L. (eds.), Handbook of Numerical Analysis, Vol. IX, North-Holland, Amsterdam, pp. 3–1176]. Like the well-known classical Fractional-Step-θ-scheme which had been introduced by Glowinski Glowinski (1985). In Murman, E. M. and Abarbanel, S. S. (eds.), Progress and Supercomputing in Computational Fluid Dynamics, Birkh?user, Boston MA; Bristeau et al. (1987). Comput. Phys. Rep. 6, 73–187], too, and which is still one of the most popular time stepping schemes, with or without operator splitting techniques, this new scheme consists of 3 substeps with nonequidistant substepping to build one macro time step. However, in contrast to the Fractional-Step-θ-scheme, the second substep can be formulated as an extrapolation step for previously computed data only, and the two remaining substeps look like a Backward Euler step so that no expensive operator evaluations for the right hand side vector with older solutions, as for instance in the Crank–Nicolson scheme, have to be performed. This modified scheme is implicit, strongly A-stable and second order accurate, too, which promises some advantageous behavior, particularly in implicit CFD simulations for the nonstationary Navier–Stokes equations. Representative numerical results, based on the software package FEATFLOW Turek (2000). FEATFLOW Finite element software for the incompressible Navier–Stokes equations: User Manual, Release 1.2, University of Dortmund] are obtained for typical flow problems with benchmark character which provide a fair rating of the solution schemes, particularly in long time simulations.Dedicated to David Gottlieb on the occasion of his 60th anniversary