Unconditionally energy gradient stable numerical scheme for Cahn–Hilliard equation with high-order degenerate mobility

In this study, we consider the Cahn–Hilliard equation with high-order degenerate mobility, where the mobility function depends on the concentration field. As a Wasserstein gradient flow of the Ginzburg–Landau free energy, the system satisfies both mass preservation and energy dissipation requirements. A novel numerical scheme based on a linear stabilized splitting method proposed satisfying mass conservation, unique solvability, and unconditional energy gradient stability while drawing connections with the porous medium equation. Extensive numerical experiments validate the theoretical findings, including mass conservation, energy stability, and accuracy, in both space and time.