Dynamically Regularized Lagrange Multiplier Method for the Cahn-Hilliard-Navier-Stokes System

This paper is concerned with efficient and accurate numerical schemes for the Cahn-Hilliard-Navier-Stokes phase field model of binary immiscible fluids. By introducing two Lagrange multipliers for each of the Cahn-Hilliard and Navier-Stokes parts, we reformulate the original model problem into an equivalent system that incorporates the energy evolution process. Such a nonlinear, coupled system is then discretized in time using first- and second-order backward differentiation formulas, in which all nonlinear terms are treated explicitly and no extra stabilization term is imposed. The proposed dynamically regularized Lagrange multiplier (DRLM) schemes are mass-conserving and unconditionally energy-stable with respect to the original variables. In addition, the schemes are fully decoupled: Each time step involves solving two biharmonic-type equations and two generalized linear Stokes systems, together with two nonlinear algebraic equations for the Lagrange multipliers. A key feature of the DRLM schemes is the introduction of the regularization parameters which ensure the unique determination of the Lagrange multipliers and mitigate the time step size constraint without affecting the accuracy of the numerical solution, especially when the interfacial width is small. Various numerical experiments are presented to illustrate the accuracy and robustness of the proposed DRLM schemes in terms of convergence, mass conservation, and energy stability.