Global error estimates of high-order fully decoupled schemes for the Cahn-Hilliard-Navier-Stokes model of Two-Phase Incompressible Flows

TL;DR

This paper introduces new high-order, fully decoupled, energy-stable schemes for two-phase incompressible flows, with rigorous error analysis and validation through numerical examples.

Contribution

It develops novel IMEX schemes based on EOP-GSAV and splitting methods, achieving optimal error estimates and unconditional energy stability.

Findings

Schemes are linear, fully decoupled, and unconditionally energy stable.

Numerical solutions are uniformly bounded without time step restrictions.

Error estimates are optimal for phase function, velocity, and pressure.

Abstract

In this paper we construct new fully decoupled and high-order implicit-explicit (IMEX) schemes for the two-phase incompressible flows based on the new generalized scalar auxiliary variable approach with optimal energy approximation (EOP-GSAV) for Cahn-Hilliard equation and consistent splitting method for Navier-Stokes equation. These schemes are linear, fully decoupled, unconditionally energy stable, only require solving a sequence of elliptic equations with constant coefficients at each time step, and provide a new technique to preserve the consistency between original energy and modified energy. We derive that numerical solutions of these schemes are uniformly bounded without any restriction on time step size. Furthermore, we carry out a rigorous error analysis for the first-order scheme and establish optimal global error estimates for the phase function, velocity and pressure in two and three-dimensional cases. Numerical examples are presented to validate the proposed schemes.