Highly efficient, robust and unconditionally energy stable second order schemes for approximating the Cahn-Hilliard-Brinkman system

Abstract

In this paper, we present an efficient, robust, and unconditionally energy stable second-order scheme for solving the Cahn-Hilliard-Brinkman (CHB) model, which mathematically describes multiphase flow in porous media. Solving the CHB model is significantly challenging due to its high coupling and nonlinearity. Here, we utilize the scalar auxiliary variable (SAV) method to handle the nonlinear term in the Cahn-Hilliard system, followed by decoupling the model through the implicit-explicit (IMEX) Crank-Nicolson approach. Our constructed scheme employs an adaptive time step, enhancing the robustness of the scheme compared to a fixed time step. The unconditional energy stability of our scheme that integrates artificial stabilized term, is assured easily owing to the SAV method and is theoretically proven in the paper. Finally, numerical experiments employing the Fourier pseudo-spectral method to discrete the spatial variables were conducted in two-dimensional space to validate the numerical accuracy and efficiency of the proposed approach.