Pressure corrected implicit θ-schemes for the incompressible Navier-Stokes equations

In this note, the Crank-Nicolson and the fractional-step theta-scheme are applied on the semi-discretised incompressible Navier-Stokes equations. In a first step, the formulation of the methods is modified such that the methods have optimal order for the pressure p, i.e. the pressure p(0) is included in the scheme. The main idea of this paper is that the fractional-step theta-scheme can be written as a diagonally implicit Runge-Kutta method (DIRK method). In this context, it can easily be shown that the fractional-step theta-schemes have only stage order q = 1 whereas the Crank-Nicolson scheme has stage order q = 2. Hence the fractional-step theta-scheme may have order reduction, if the method is applied on stiff ODEs and DAEs, i.e. the semi-discretised incompressible Navier-Stokes equations. Some theoretical results and numerical examples illustrate this phenomena. Moreover, it is shown that it is impossible to improve the fractional-step- theta-method such that the scheme has the stage order q 2 and is strongly A-stable or has the order p = 3.