Implicit-explicit all-speed schemes for compressible Cahn-Hilliard-Navier-Stokes equations
We propose a second-order implicit-explicit (IMEX) time-stepping scheme for the isentropic, compressible Cahn-Hilliard-Navier-Stokes equations in the low Mach number regime. The method is based on finite differences on staggered grids and is specifically designed to handle the challenges posed by the low Mach number limit, where the system approaches to an incompressible behavior. In this regime, standard explicit schemes suffer from severe time-step restrictions due to fourth-order diffusion terms and the stiffness induced by fast acoustic waves. To overcome this, we employ an IMEX strategy which splits the governing equations into stiff and non-stiff components. The stiff terms, arising from pressure, viscous forces and fourth-order Cahn-Hilliard contributions, are treated implicitly, while the remaining are dealt explicitly.