Convergence of a Decoupled Splitting Scheme for the Cahn-Hilliard-Navier-Stokes System

TL;DR

This paper introduces an energy-stable, decoupled splitting scheme for the Cahn-Hilliard-Navier-Stokes system, providing rigorous stability, solvability, and error estimates without regularization.

Contribution

It develops a novel discontinuous Galerkin algorithm that is energy-stable, uniquely solvable, mass conservative, and offers optimal error estimates for the coupled system.

Findings

Scheme is energy-stable and mass conservative.

Achieves optimal error estimates in relevant norms.

Ensures stability without regularizing the potential function.

Abstract

This paper is devoted to the analysis of an energy-stable discontinuous Galerkin algorithm for solving the Cahn-Hilliard-Navier-Stokes equations within a decoupled splitting framework. We show that the proposed scheme is uniquely solvable and mass conservative. The energy dissipation and the $L^\infty$ stability of the order parameter are obtained under a CFL condition. Optimal a priori error estimates in the broken gradient norm and in the $L^2$ norm are derived. The stability proofs and error analysis are based on induction arguments and do not require any regularization of the potential function.