Among the most popular methods for solving Navier-Stokes Equation, the penalty-projection is known to combine the advantages of the projection method and the penalty method to design an accurate and efficient scheme.
As the projection methods, they use a splitting between the nonlinear equation and the projection onto the velocity space with free divergence. The idea of the penalty-projection methods is to add to the nonlinear viscous step a penalty term, "improving" the intermediate velocity. Although some of them have been designed to provide an accurate rate of convergence, their implementation can involve a prohibitive cost.
In this talk, we introduce a variant of the penalty-projection method that combines dynamically and alternatively a penalty procedure and a projection procedure according to the size of the divergence of the velocity. Theoretical estimates for the new method are given, which are in accordance with the numerical results provided.