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This is a Quantitative Study study.

Optimal convergence of a fully decoupled finite difference scheme of the Abels–Garcke–Grün model for incompressible two-phase flows with unmatched densities

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Novel finite difference scheme shows optimal convergence for incompressible two-phase flows, indicating stability and error characteristics.

The proposed scheme guarantees the unique solvability and unconditional energy stability of the model.
It maintains positivity in the phase-field variable, ensuring solutions remain valid within the physical interval.
The convergence analysis reveals first-order accuracy in time and second-order accuracy in space for error estimates.
Numerical experiments support the theoretical findings, demonstrating the scheme's effectiveness for critical phase-field applications.

Chen et al. (2025) studied this question.

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Novel finite difference scheme shows optimal convergence for incompressible two-phase flows, indicating stability and error characteristics.