Solver methods for nonlinear diffusion equation: Demixing of two species

We consider two species obeying nonlinear diffusion equations. Therein, the diffusion coefficient of a each species increases locally and nonlinearly with the density of the second species. This simple assumption originates demixing of the two species if the dependence on the second density becomes over critically strong. The demixing is characterized by sequences of separated regions with higher densities of one species compared to the second one. Since the model does not contain a nonlocal interaction no interface energy is created and the density profile jumps abruptly between large values at particular positions. In this manuscript, we describe numerically this stiff behavior. We propose a conservative discretization scheme in this paper, which earlier was also applied to Cahn-Hilliard and nonlinear Diffusion-Reaction equations. We modify the scheme with respect to the demixture model and find numerically stationary stable spatial distributions. Several aspects of the model and its treatment are discussed.