Implicit vectorial operator splitting for incompressible Navier-Stokes equations in primitive variables (Q2773661)

The authors consider steady flows of incompressible viscous fluid in rectangular cavities. The Navier-Stokes equations and incompressibility constraint constitute an elliptic system which is combined with a Poisson-type equation for pressure instead of continuity equation. Unlike the classical Poisson equation for pressure, the equation derived here contains the original continuity equation as an additional term. This proves to be essential for stability of the quasi-stationary method and for the application of operator splitting. The authors have shown that the new system is equivalent to the original Navier-Stokes system provided that the continuity condition is imposed as an additional boundary condition, and no boundary conditions for pressure are specified. A novel and computationally efficient vectorial implicit operator-splitting method for solving the evolution system is developed. The nonlinear terms are approximated by conservative central differences. Although devoid of artificial (``scheme'') viscosity, the scheme is shown to be stable for reasonable time steps even at very high Reynolds numbers. The consistency and convergence of the scheme are shown analytically and verified numerically via benchmark tests with different resolutions and time steps.