An Analysis on the Finite Volume Schemes and the Discrete Lyapunov Inequalities for the Chemotaxis System

Abstract

We analyze two finite volume schemes, linear and nonlinear, for the chemotaxis system in two-dimensional domain, which preserve the mass conservation and positivity without the CFL condition. For the nonlinear scheme, the well-posedness is proved by using Brouwer’s fixed point theory, and we show the convergence of the Picard iteration. We also investigate two discrete Lyapunov functionals, the asymptotic stability of equilibrium and the local stability. Moreover, we apply the discrete semi-group theory to error analysis and obtain the convergence rate $$O(\tau +h)$$ in $$L^p$$ norm. The theoretical results are confirmed by numerical experiments.