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

Abstract

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 ?-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-?-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, Birkhauser, 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-?-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.