Convergence analysis of a second order numerical scheme for the Flory-Huggins-Cahn-Hilliard-Navier-Stokes system
Wenbin Chen, Jianyu Jing, Qianqian Liu, Cheng Wang, Xiaoming Wang
TL;DR
This work delivers the first rigorous proof of optimal second-order convergence in both time and space for a fully discrete CHNS system with a logarithmic Flory–Huggins energy potential. The authors design a second-order Crank–Nicolson–type scheme on a MAC grid, incorporating a nonlinear regularization to preserve the positivity of the logarithmic terms and an energy-stable formulation. The convergence analysis combines a higher-order consistency framework, a rough error bound, and a refined nonlinear error estimate, leveraging a discrete Helmholtz projection to maintain divergence-free velocity fields and a careful treatment of nonlinear logarithmic terms. The results provide theoretical guarantees for accuracy and stability that are crucial for reliable long-time simulations of phase-field–fluid interactions with singular energy potentials, and they establish a methodological blueprint for analyzing similar coupled nonlinear PDE systems.
Abstract
We present an optimal rate convergence analysis for a second order accurate in time, fully discrete finite difference scheme for the Cahn-Hilliard-Navier-Stokes (CHNS) system, combined with logarithmic Flory-Huggins energy potential. The numerical scheme has been recently proposed, and the positivity-preserving property of the logarithmic arguments, as well as the total energy stability, have been theoretically justified. In this paper, we rigorously prove second order convergence of the proposed numerical scheme, in both time and space. Since the CHNS is a coupled system, the standard $\ell^\infty (0, T; \ell^2) \cap \ell^2 (0, T; H_h^2)$ error estimate could not be easily derived, due to the lack of regularity to control the numerical error associated with the coupled terms. Instead, the $\ell^\infty (0, T; H_h^1) \cap \ell^2 (0, T; H_h^3)$ error analysis for the phase variable and the $\ell^\infty (0, T; \ell^2)$ analysis for the velocity vector, which shares the same regularity as the energy estimate, is more suitable to pass through the nonlinear analysis for the error terms associated with the coupled physical process. Furthermore, the highly nonlinear and singular nature of the logarithmic error terms makes the convergence analysis even more challenging, since a uniform distance between the numerical solution and the singular limit values of is needed for the associated error estimate. Many highly non-standard estimates, such as a higher order asymptotic expansion of the numerical solution (up to the third order accuracy in time and fourth order in space), combined with a rough error estimate (to establish the maximum norm bound for the phase variable), as well as a refined error estimate, have to be carried out to conclude the desired convergence result.