A Positivity-Preserving, Energy Stable BDF2 Scheme with Variable Steps for the Cahn–Hilliard Equation with Logarithmic Potential

We propose and analyze a BDF2 scheme with variable time steps for the Cahn–Hilliard equation with a logarithmic Flory–Huggins energy potential. The lumped mass method is adopted in the space discretization to ensure that the proposed scheme is uniquely solvable and positivity-preserving. Especially, a new second order viscous regularization term is added at the discrete level to guarantee the energy dissipation property. Furthermore, the energy stability is derived by a careful estimate under the condition that \(r\le r_{\max }\). To estimate the spatial and temporal errors separately, a spatially semi-discrete scheme is proposed and a new elliptic projection is introduced, and the super-closeness between this projection and the Ritz projection of the exact solution is attained. Based on the strict separation property of the numerical solution obtained by using the technique of combining the rough and refined error estimates, the convergence analysis in \(l^{\infty }(0,T;L_h^2(\varOmega ))\) norm is established when \(\tau \le Ch\) by using the technique of the DOC kernels. Finally, several numerical experiments are carried out to validate the theoretical results.