A consistent viscosity-splitting method for the Navier–Stokes equations with pressure boundary conditions

The incompressible Navier–Stokes equations equipped with classical boundary conditions have been extensively studied from both theoretical and numerical perspectives, and a wide variety of fractional-step and splitting methods have been developed to reduce their computational complexity. However, in many realistic flow configurations, standard boundary conditions are inadequate to accurately represent the physical behavior at open or artificial boundaries, where mixed or pressure-based conditions are more appropriate. Despite their practical relevance, such unconventional boundary conditions have received comparatively little attention in the design and analysis of consistent splitting schemes. In the present work, we propose a class of viscosity-splitting schemes for the incompressible Navier–Stokes equations subject to non-standard boundary conditions. In particular, the velocity is prescribed in a Dirichlet sense on a portion of the boundary, while on the complementary part the tangential velocity together with the total pressure is imposed. The proposed schemes are constructed to ensure consistency with the continuous problem and to properly account for prescribed pressure on the boundary. To enhance mass conservation and improve numerical robustness, a grad-div stabilization term is incorporated into the formulation. The convergence and accuracy properties of the resulting methods are assessed using a series of numerical examples, which demonstrate their effectiveness in handling pressure boundary conditions and confirm the expected convergence behavior. These results indicate that the proposed viscosity-splitting approach provides a reliable and efficient method for the simulation of incompressible flows with physically relevant, non-standard boundary conditions.