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

In this note, the Crank–Nicolson and the fractional-step θθ-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 p0p0 is included in the scheme.The main idea of this paper is that the fractional-step θθ-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 θθ-schemes have only stage order q=1q=1 whereas the Crank–Nicolson scheme has stage order q=2q=2. Hence the fractional-step θθ-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-θθ-method such that the scheme has the stage order q=2q=2 and is strongly A-stable or has the order p=3p=3.