Error estimates of linear decoupled structure-preserving incremental viscosity splitting methods for the Cahn--Hilliard--Navier--Stokes system

TL;DR

This paper introduces efficient, linear, decoupled time discretization schemes for the coupled Cahn--Hilliard--Navier--Stokes system, with proven stability, mass conservation, and optimal error convergence in multiple dimensions.

Contribution

It develops novel first- and second-order schemes using incremental viscosity splitting that are unconditionally energy dissipative and rigorously analyzed for error bounds.

Findings

Schemes are uniquely solvable and mass-conserving.

Proven unconditional energy dissipation and optimal convergence rates.

Numerical tests confirm theoretical accuracy and efficiency.

Abstract

We propose first- and second-order time discretization schemes for the coupled Cahn--Hilliard--Navier--Stokes model, leveraging the incremental viscosity splitting (IVS) method. The schemes combine the scalar auxiliary variable method and the zero-energy-contribution approach, resulting in a linear, decoupled numerical framework. At each time step, they only require to solve a sequence of constant-coefficient equations, along with a linear equation with one unknown, making the algorithms computationally efficient and easy to implement. In addition, the proposed schemes are proven to be uniquely solvable, mass-conserving, and unconditional energy dissipation. Most importantly, leveraging the mathematical induction method and the regularity properties of the Stokes equation, we perform a rigorous error analysis for the first-order scheme in multiple space dimensions, establishing an unconditional and optimal convergence rate for all relevant variables under different norms. A user-defined, time-dependent parameter plays an important role in the error analysis of the proposed structure-preserving IVS methods. Ample numerical examples are carried out to verify the theoretical findings and to demonstrate the accuracy, effectiveness and efficiency of the proposed schemes.