An implicit monolithic time accurate finite element scheme for incompressible flow problems

Abstract

The objective of this paper is to develop an implicit, monolithic, finite element (FE) scheme for the solution of the incompressible Navier-Stokes (NS) equations. The design of the method is based on the pressure stability properties of an implicit second order in time fractional step (FS) method, which is conditionally stable. The final monolithic scheme preserves the second order accuracy of the FS method, and it is unconditionally stable (i.e. stable for all timestep sizes). In addition, it is shown that the final pressure stabilizing term is practically the same fourth order pressure term added by some authors, but following different arguments, to obtain high order accurate results. It is also shown that the final stabilized convective term is a formally second order discretization of the advective operator. Finally, a non-linear numerical switch was designed to lower the discretization order in flow regions where the solution could present over or undershoots, which can inhibit the algorithm convergence. Some numerical examples are presented.