Long time stability of a classical efficient scheme for an incompressible two-phase flow model

TL;DR

This paper proves that the implicit Euler scheme for a 2D two-phase flow model has global attractors that converge to the continuous system's attractor as the time-step decreases, ensuring long-term stability.

Contribution

It establishes the convergence of the numerical scheme's global attractors to the continuous system's attractor using discrete Gronwall lemmas.

Findings

Global attractors of the numerical scheme converge to the continuous system's attractor.

The implicit Euler scheme is stable over long time periods.

Mathematical proof using discrete Gronwall lemmas.

Abstract

In this article we consider the implicit Euler scheme for a homogeneous two-phase flow model in a two-dimensional domain and with the aid of the discrete Gronwall lemma and of the discrete uniform Gronwall lemma we prove that the global attractors generated by the numerical scheme converge to the global attractor of the continuous system as the time-step approaches zero.