A stable and linear time discretization for a thermodynamically consistent model for two-phase incompressible flow

TL;DR

This paper introduces a new time discretization scheme for two-phase flow models that ensures energy stability and includes an adaptive spatial discretization with error estimation, improving simulation accuracy and reliability.

Contribution

It presents a thermodynamically consistent, energy-stable discretization scheme with adaptive spatial refinement and error estimation for two-phase flow simulations.

Findings

The scheme maintains a discrete energy inequality.

Adaptive spatial discretization conserves energy in the fully discrete setting.

A quasi-reliable error estimator effectively estimates flow and phase field errors.

Abstract

A new time discretization scheme for the numerical simulation of two-phase flow governed by a thermodynamically consistent diffuse interface model is presented. The scheme is consistent in the sense that it allows for a discrete in time energy inequality. An adaptive spatial discretization is proposed that conserves the energy inequality in the fully discrete setting by applying a suitable post processing step to the adaptive cycle. For the fully discrete scheme a quasi-reliable error estimator is derived which estimates the error both of the flow velocity, and of the phase field. The validity of the energy inequality in the fully discrete setting is numerically investigated.