Unconditionally energy stable Cahn-Hilliard equation using adaptive potential functions

The Cahn-Hilliard equation is fundamental for modelling phase separation and coarsening in binary mixtures. Owing to its stiffness, finding an exact solution is difficult. The choice of free energy potential strongly influences the model's physical accuracy and numerical stability. The logarithmic potential accurately captures thermodynamic singularities at pure phases and suffers from severe stiffness near extreme concentrations. Conversely, the polynomial potential is more stable and computationally efficient but less physically realistic. We propose an adaptive potential approach that switches pointwise between logarithmic and polynomial forms depending on concentration values relative to a tolerance parameter. The time evolution is solved using Eyre's semi-implicit splitting scheme, which ensures unconditional energy stability by treating nonlinear terms explicitly and linear terms implicitly with first- and second-order backward differentiation. Spatial discretization employs a Fourier cosine spectral method with Neumann boundaries, ensuring mass conservation and monotonic energy dissipation.