CONVERGENCE ANALYSIS OF A DISCRETE DUALITY FINITE-VOLUME METHOD FOR CAHN-HILLIARD EQUATIONS WITH SURFACTANTS
Abstract
We propose a Discrete Duality Finite Volume (DDFV) scheme for a phase-field model of surfactant dynamics during the phase separation of a two-phase flow. The model couples two Cahn-Hilliard equations, for the phase-field variable and for the surfactant concentration. We discretize the model in space via the DDFV method, well-suited to general meshes and nonlinear diffusion, and in time via a semi-implicit Euler scheme. The discrete scheme and the corresponding numerical analysis rely on two dedicated operators discretizing the nonlinear term coupling the surfactant concentration and the phase-field gradient consistently across the primal, dual and diamond meshes. We first prove the existence of a discrete solution at each time step, via a topological degree argument. We then prove that, as the mesh size and time step tend to zero, the approximate solutions converge to a weak solution of the continuous problem. The existence of a solution to the continuous problem then follows as a consequence. To our knowledge, the finite volume analysis, the convergence of the scheme and the existence of a continuous solution are all new for this model. We finally report numerical simulations illustrating the interface-driven surfactant dynamics, including a benchmark showing the stabilization of the interfaces under high surfactant concentration.
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BibTeX
@misc{castellano2026convergence,
title = {{CONVERGENCE ANALYSIS OF A DISCRETE DUALITY FINITE-VOLUME METHOD FOR CAHN-HILLIARD EQUATIONS WITH SURFACTANTS}},
author = {Margherita Castellano and Ludovic Goudenège and Flore Nabet},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05747365v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - CONVERGENCE ANALYSIS OF A DISCRETE DUALITY FINITE-VOLUME METHOD FOR CAHN-HILLIARD EQUATIONS WITH SURFACTANTS
AU - Margherita Castellano
AU - Ludovic Goudenège
AU - Flore Nabet
PY - 2026
DA - 2026/09/11
PB - HAL
UR - https://hal.science/hal-05747365v1
AB - We propose a Discrete Duality Finite Volume (DDFV) scheme for a phase-field model of surfactant dynamics during the phase separation of a two-phase flow. The model couples two Cahn-Hilliard equations, for the phase-field variable and for the surfactant concentration. We discretize the model in space via the DDFV method, well-suited to general meshes and nonlinear diffusion, and in time via a semi-implicit Euler scheme. The discrete scheme and the corresponding numerical analysis rely on two dedicated operators discretizing the nonlinear term coupling the surfactant concentration and the phase-field gradient consistently across the primal, dual and diamond meshes. We first prove the existence of a discrete solution at each time step, via a topological degree argument. We then prove that, as the mesh size and time step tend to zero, the approximate solutions converge to a weak solution of the continuous problem. The existence of a solution to the continuous problem then follows as a consequence. To our knowledge, the finite volume analysis, the convergence of the scheme and the existence of a continuous solution are all new for this model. We finally report numerical simulations illustrating the interface-driven surfactant dynamics, including a benchmark showing the stabilization of the interfaces under high surfactant concentration.
ER -