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The present paper deals with the numerical solution of the incompressibleNavier-Stokes equations using high-order discontinuous Galerkin (DG) methodsfor discretization in space. For DG methods applied to the dual splittingprojection method, instabilities have recently been reported that occur forcoarse spatial resolutions and small time step sizes. By means of numericalinvestigation we give evidence that these instabilities are related to thediscontinuous Galerkin formulation of the velocity divergence term and thepressure gradient term that couple velocity and pressure. Integration by partsof these terms with a suitable definition of boundary conditions is required inorder to obtain a stable and robust method. Since the intermediate velocityfield does not fulfill the boundary conditions prescribed for the velocity, aconsistent boundary condition is derived from the convective step of the dualsplitting scheme to ensure high-order accuracy with respect to the temporaldiscretization. This new formulation is stable in the limit of small time stepsfor both equal-order and mixed-order polynomial approximations. Although thedual splitting scheme itself includes inf-sup stabilizing contributions, wedemonstrate that spurious pressure oscillations appear for equal-orderpolynomials and small time steps highlighting the necessity to consider inf-supstability explicitly.

On the stability of projection methods for the incompressible Navier-Stokes equations based on high-order discontinuous Galerkin discretizations