A second-order fully-balanced structure-preserving variational discretization scheme for the Cahn-Hilliard Navier-Stokes system

TL;DR

This paper introduces a second-order, structure-preserving variational discretization scheme for the Cahn-Hilliard-Navier-Stokes system, ensuring stability, convergence, and practical applicability with demonstrated numerical results.

Contribution

It develops a novel second-order, fully-balanced discretization method that preserves the system's structure and stability, with proven convergence and applicability to complex parameters.

Findings

Establishes stability and uniqueness of the discrete scheme.

Achieves order optimal convergence rates for all variables.

Numerical tests confirm practical effectiveness and predicted convergence.

Abstract

We propose and analyze a structure-preserving space-time variational discretization method for the Cahn-Hilliard-Navier-Stokes system. Uniqueness and stability for the discrete problem is established in the presence of concentration dependent mobility and viscosity parameters by means of the relative energy estimates and order optimal convergence rates are established for all variables using balanced approximation spaces and relaxed regularity conditions on the solution. Numerical tests are presented to demonstrate the proposed method is fully practical and yields the predicted convergence rates. The discrete stability estimates developed in this paper may also be used to analyse other discretization schemes, which is briefly outlined in the discussion.