A discussion of artificial compressibility

The artificial compressibility method for solving the unsteady Navier–Stokes equations is discussed. Typically, this method is combined with a dual time stepping procedure which involves an iteration in pseudo-time within each physical time step, and which guarantees a convergence towards the solution for the incompressible flow problem.

Numerous examples from the literature, however, ignore the pseudo-time iteration, in which case the choice of artificial compressibility becomes a tradeoff between accuracy and numerical efficiency. In this case it becomes important to establish thresholds for the artificial compressibility. In order to establish such thresholds, we derive linear wave solutions for progressive and standing waves under the influence of artificial compressibility.

Thresholds are given in terms of the Mach number and the Courant number.