A penalty-projection method for dilatable flows

We present in this paper a novel pressure correction method for dilatable flows. Named "penalty-projection method", this scheme differs from the usual projection method by the addition in the prediction step of a penalty term, built to enforce the mass balance. This term is proportional to a penalty parameter r. Numerical experiments show that the accuracy of the scheme is drastically improved. The splitting error, dominant at large time steps, may be reduced down to zero by choosing larger and larger values for r; note, however, that a too large value degrades the conditioning of the operator associated to the prediction step. In addition, the loss of convergence observed with the standard projection method in case of open boundary conditions is not observed anymore, as soon as r is nonzero.