Numerical analysis of a new time-stepping θ-scheme for incompressible flow simulations

In [13], 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 new time-stepping θ-scheme which has been recently proposed by Glowinski [5]. Like the well-known classical FractionalStep-θ-scheme which had been introduced by Glowinski [1], [4], 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 non-equidistant 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 new 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 [15], are obtained for typical flow problems with benchmark character which provide a fair rating of the solution schemes, particularly in long time simulations.