Second-order linear energy-stable relaxed Lagrange multiplier (RLM) schemes for Cahn--Hilliard--Navier--Stokes phase-field models

Abstract

We propose a new class of second-order, linear, unconditionally energy-stable schemes for the Cahn--Hilliard--Navier--Stokes (CHNS) system. A single scalar Lagrange multiplier is introduced to enforce the discrete energy dissipation law for the coupled system. At each time step, the phase-field variables, chemical potential, and velocity are obtained by solving four independent linear equations, after which the multiplier is updated via a single scalar quadratic equation with an explicit closed-form solution. This contrasts with other existing numerical schemes for the CHNS system. We provide rigorous proofs that unconditional energy stability holds for the proposed schemes. Numerical experiments are then presented to confirm second-order temporal accuracy. Compared with other existing numerical schemes for the CHNS system, our schemes achieve better or comparable accuracy and efficiency.