A discrete boundedness-by-entropy method for finite-volume approximations of cross-diffusion systems

Abstract An implicit Euler finite-volume scheme for general cross-diffusion systems with volume-filling constraints is proposed and analyzed. The diffusion matrix may be nonsymmetric not positive semidefinite, but the system assumed to possess a formal gradient-flow structure that yields $L^\infty $ bounds on continuous level. Examples include Maxwell–Stefan gas mixtures, tumor-growth models fabrication of thin-film solar cells. numerical preserves equations, namely entropy dissipation inequality as well non-negativity concentrations constraints. discrete consequence new vector-valued chain rule. existence solutions, their positivity, convergence proved. implemented one-dimensional model two-dimensional cell system. It illustrated rate in space order two relative decays exponentially.