A simple and efficient finite difference scheme to the Cahn–Hilliard–Navier–Stokes system equations

Central difference schemes are usually not suited to convective terms in a transport equation due to its oscillation nature when convection effect is pronounced. However, this work found that applying the standard central difference scheme to the convective term along with another central difference scheme to the fourth order diffusive term in the Cahn–Hilliard equation can realize nearly an order of magnitude speed rise, in the framework of a fully explicit finite difference scheme. The discretization was done on a semi-staggered grid where pressure was stored at the cell center and other variables were stored at the cell corners. To accelerate computation, a simple parallelism based on OpenMP was used. The scheme was tested in a number of cases and was compared with both analytical and experimental outcomes. The results showed that the scheme is efficient compared with a previous fully explicit finite difference scheme for the Cahn–Hilliard equation, and that a time step more than five times larger can be employed.