Implicit-explicit schemes for compressible Cahn-Hilliard-Navier-Stokes equations on staggered grids

TL;DR

This paper introduces a second-order IMEX scheme for simulating compressible Cahn-Hilliard-Navier-Stokes equations on staggered grids, improving stability and efficiency over explicit methods.

Contribution

The paper develops a novel second-order IMEX Runge-Kutta scheme that handles stiff terms implicitly and convective terms explicitly for compressible multiphase flow equations.

Findings

Scheme is stable and accurate in numerical tests

Reduces computational cost compared to explicit methods

Effective for stiff, high-order diffusion problems

Abstract

We propose a second-order implicit-explicit (IMEX) time-stepping scheme for the isentropic, compressible Cahn-Hilliard-Navier-Stokes equations discretized on staggered (MAC) grids. The scheme is based on finite difference approximations that ensure a stable coupling among the velocity, density and phase field, with symmetric operators acting on the discretized viscosity terms. Standard explicit methods suffer from severe time-step restrictions due to the presence of second to fourth-order diffusion terms introduced by the Cahn-Hilliard and Navier-Stokes operators. To overcome these challenges, we develop an IMEX Runge-Kutta scheme that treats the stiff terms implicitly while the convective terms are dealt with explicitly, with the advantage that only linear systems are solved at each stage. Numerical experiments are performed to verify the stability, accuracy and efficiency of the proposed approach.