Implicit-explicit all-speed schemes for compressible Cahn-Hilliard-Navier-Stokes equations

TL;DR

This paper introduces a second-order IMEX scheme for simulating compressible Cahn-Hilliard-Navier-Stokes equations at low Mach numbers, effectively handling stiffness and enabling larger time steps.

Contribution

It develops a novel IMEX time-stepping method tailored for low Mach number regimes, improving stability and efficiency over explicit schemes.

Findings

The scheme achieves second-order accuracy in time.

It effectively handles stiffness from fourth-order diffusion and acoustic waves.

The method allows larger time steps without sacrificing stability.

Abstract

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.