A new approach for the numerical solution of diffusion equations with variable and degenerate mobility

We propose a novel approach for the numerical integration of diffusion-type equations with variable and degenerate mobility or diffusion coefficient. Our focus is the Cahn-Hilliard equation which plays a prominent role in phase field models of fluids and soft materials but the methodology has a more general applicability. The central idea is a split method with a linearly implicit component and an analytic step to integrate out the variable mobility. The proposed method is robust, free of high order stability constraints, and its cost is comparable to that of solving the linear Heat Equation with the backward Euler Method. Moreover, by design, the numerical solution is guaranteed to be strictly bounded by the stable, constant states.