A class of IMEX schemes and their error analysis for the Navier–Stokes Cahn–Hilliard system

We construct a class of implicit-explicit (IMEX) schemes for the Navier–Stokes Cahn–Hilliard (NSCH) system and carry out a rigorous error analysis for both semi-discrete and fully discrete (with a Fourier spectral approximation in space) schemes in the space periodic case. The schemes are based on the consistent splitting approach for the Navier–Stokes equations to decouple the computation of velocity and pressure, and the generalized scalar auxiliary variable (GSAV) approach to provide uniform bound for the numerical solutions. Our IMEX schemes are fully decoupled and linear, only requiring to solve a sequence of Poisson type equations at each time step. With help of the uniform bound for the numerical solutions, we derive global error estimates in the two-dimensional case as well as local error estimates in the three-dimensional case for temporal orders one to five. We also present some numerical examples to validate the schemes.

Keywords

two-phase flow, Navier–Stokes, Cahn–Hilliard, stability, error analysis, high-order, IMEX

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Dedicated to the memory of Professor Roland Glowinski

This work was supported in part by NSFC 12371409.

Received 28 January 2023

Accepted 10 February 2024

Published 5 April 2024