New highly efficient and accurate numerical scheme for the Cahn-Hilliard-Brinkman system

In this paper, based on a generalized scalar auxiliary variable approach with relaxation (R-GSAV), we construct a class of high-order backward differentiation formula (BDF) schemes with variable time steps for the Cahn-Hilliard-Brinkman(CHB) system. In theory, it is strictly proved that the designed schemes are unconditionally energy-stable. With the delicate treatment of adaptive strategies, we propose several adaptive time-step algorithms to enhance the robustness of the schemes. More importantly, a novel hybrid-order adaptive time steps algorithm performs outstanding for the coupled system. The hybrid-order algorithm inherits the advantages of some traditional high-order BDF adaptive strategies. A comprehensive comparison with some adaptive time-step algorithms is given, and the advantages of the new adaptive time-step algorithms are emphasized. Finally, the effectiveness and accuracy of the new methods are validated through a series of numerical experiments.