A projection-based, semi-implicit time-stepping approach for the Cahn-Hilliard Navier-Stokes equations on adaptive octree meshes

TL;DR

This paper introduces a semi-implicit, projection-based finite element method for efficiently solving the coupled Cahn-Hilliard Navier-Stokes equations on adaptive octree meshes, enabling larger time steps and faster computations in two-phase fluid flow simulations.

Contribution

It presents a novel semi-implicit, projection-based finite element framework with scalable AMG solvers and adaptive meshes for the CHNS system, improving computational efficiency and stability.

Findings

Faster time-to-solve compared to fully-implicit methods.

Accurate simulation of bubble rise and Rayleigh-Taylor instability.

Efficient parallel implementation on adaptive octree meshes.

Abstract

The Cahn-Hilliard Navier-Stokes (CHNS) system provides a computationally tractable model that can be used to effectively capture interfacial dynamics in two-phase fluid flows. In this work, we present a semi-implicit, projection-based finite element framework for solving the CHNS system. We use a projection-based semi-implicit time discretization for the Navier-Stokes equation and a fully-implicit time discretization for the Cahn-Hilliard equation. We use a conforming continuous Galerkin (cG) finite element method in space equipped with a residual-based variational multiscale (RBVMS) formulation. Pressure is decoupled using a projection step, which results in two linear positive semi-definite systems for velocity and pressure, instead of the saddle point system of a pressure-stabilized method. All the linear systems are solved using an efficient and scalable algebraic multigrid (AMG) method. We deploy this approach on a massively parallel numerical implementation using parallel octree-based adaptive meshes. The overall approach allows the use of relatively large time steps with much faster time-to-solve than similar fully-implicit methods. We present comprehensive numerical experiments showing detailed comparisons with results from the literature for canonical cases, including the single bubble rise and Rayleigh-Taylor instability.