Positivity-preserving, energy stable numerical schemes for the Cahn-Hilliard equation with logarithmic potential

Abstract

In this paper we present and analyze finite difference numerical schemes for the Cahn-Hilliard equation with a logarithmic Flory Huggins energy potential. Both first and second order accurate temporal algorithms are considered. In the first order scheme, we treat the nonlinear logarithmic terms and the surface diffusion term implicitly, and update the linear expansive term and the mobility explicitly. We provide a theoretical justification that this numerical algorithm has a unique solution, such that the positivity is always preserved for the logarithmic arguments, i.e., the phase variable is always between −1 and 1, at a point-wise level. In particular, our analysis reveals a subtle fact: the singular nature of the logarithmic term around the values of −1 and 1 prevents the numerical solution reaching these singular values, so that the numerical scheme is always well-defined as long as the numerical solution stays similarly bounded at the previous time step. Furthermore, an unconditional energy stability of the numerical scheme is derived, without any restriction for the time step size. Such an analysis technique can also be applied to a second order numerical scheme in which the BDF temporal stencil is applied, the expansive term is updated by a second order Adams-Bashforth explicit extrapolation formula, and an artificial Douglas-Dupont regularization term is added to ensure the energy dissipativity. The unique solvability and the positivity-preserving property for the second order scheme are proved using similar ideas, namely, the singular nature of the logarithmic term plays an essential role. For both the first and second order accurate schemes, we are able to derive an optimal rate convergence analysis. The case with a non-constant mobility is analyzed as well. We also describe a practical and efficient multigrid solver for the proposed numerical schemes, and present some numerical results, which demonstrate the robustness of the numerical schemes.