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

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 Lp norm. The theoretical results are confirmed by numerical experiments.

Automated summary

In this study, we analyze two finite volume schemes for the chemotaxis system in a two-dimensional domain, demonstrating mass conservation, positivity, and well-posedness without the need for the CFL condition. We investigate the stability of equilibrium and local stability, and apply discrete semigroup theory for error analysis, achieving a convergence rate O(tau+h) in Lp norm. The theoretical results are validated through numerical experiments.