A Multiple Scalar Auxiliary Variables Approach to the Energy Stable Scheme of the Moving Contact Line Problem

In this article, we explore the numerical approximation for a phase-field model to the moving contact line (MCL) problem, governed by the Cahn-Hilliard equation with a dynamic contact angle condition. A difficult and challenging issue comes from the high order, nonlinear and coupled time marching system with a small dimensionless parameter describing the interface thickness. The scalar auxiliary variable (SAV) approach has been recently developed for a large class of gradient flows in principle. Due to the nonlinear gradient terms in both the bulk and boundary energies, we introduce multiple scalar auxiliary variables (MSAV) approach rather than single SAV scheme to deal with phase field contact line model. The MSAV scheme numerically gives a better description of the contact line dynamic in terms of the difference between the modified and original energies. Moreover, the resulting numerical schemes enjoy the advantages of second order accuracy, unconditional energy stability, and only require solving a linear decoupled system with constant coefficients at each time step. Numerical experiments are presented to verify the effectiveness and efficiency of the proposed schemes for a wide range of mobility parameters and phenomenological boundary relaxation parameters.