A Positivity‐Preserving, Energy Stable BDF2 Scheme With Variable Time Steps for the Liquid Thin Film Coarsening Model

ABSTRACT We present and analyze a second‐order numerical scheme with variable time steps for a liquid thin film coarsening model, which is a Cahn–Hilliard‐type equation with a singular Leonard–Jones energy potential. The fully discrete scheme is mainly based on the Backward differentiation formula (BDF) method in time derivation combined with the finite difference method in spacial discretization. A second‐order viscous regularization term is added at the discrete level to guarantee the energy dissipation property under the condition that the ratio of adjacent time steps is no more than an artificial constant . The uniquely solvable and positivity‐preserving properties of the numerical solution are established at a theoretical level. In addition, based on the strict separation property of the numerical solution obtained by using the technique of combining the rough and refined error estimates, the optimal rate convergence analysis is established by using the technique of the discrete orthogonal convolution (DOC) kernels when the maximum time step and space step satisfy a linear constraint . Finally, several numerical experiments are carried out to validate the theoretical results.