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

TL;DR

The paper tackles efficient simulation of the isentropic compressible Cahn–Hilliard–Navier–Stokes system with gravity, a challenging fourth‑order PDE model for binary fluids. It introduces a second‑order linearly implicit–explicit IMEX time‑stepping method within a method‑of‑lines framework, enabling only linear solves by treating convection explicitly and stiff CHNS terms implicitly. The main contributions are the design of IMEX schemes (including a second‑order *‑DIRKSA method), a structured solver pipeline with SPD linear systems, and comprehensive 2D numerical experiments that demonstrate second‑order temporal accuracy under convective CFL restrictions and favorable solver performance. The work lays a foundation for efficient, scalable simulations of diffuse‑interface fluid mixtures and points to extensions to 3D, stiffer equations, and quasi‑incompressible models.

Abstract

The isentropic compressible Cahn-Hilliard-Navier-Stokes equations is a system of fourth-order partial differential equations that model the evolution of some binary fluids under convection. The purpose of this paper is the design of efficient numerical schemes to approximate the solution of initial-boundary value problems with these equations. The efficiency stems from the implicit treatment of the high-order terms in the equations. Our proposal is a second-order linearly implicit-explicit time stepping scheme applied in a method of lines approach, in which the convective terms are treated explicitly and only linear systems have to be solved. Some experiments are performed to assess the validity and efficiency of this proposal.

Figures (11)

• Figure 1: Conservation errors for test 3.
• Figure 2: Test 1. Left: Time evolution of the CFL parameter; Right: Time evolution of $\min c$, $\max c$.
• Figure 3: Results for Test 1. Left: Initial condition, with $c$-variable inside spinodal region; Right: Results for $T=0.1$, where density increases at the bottom and separation is clearly visible in the $c$-variable.
• Figure 4: Results for Test 1, $T=0.3$ (left) and $T=0.5$ (right) where it can be seen that density continues increasing at the bottom, forming an upgoing front, and nucleation is beginning, as seen in the $c$-variable.
• Figure 5: Results for Test1, $T=0.7$ (left) and $T=1.0$ (right) where it can be seen in the velocity that vorticity has developed and nucleation is increasing, as seen in the $c$-variable.