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.

我们分析了二维域中趋化系统的两​​种线性和非线性有限体积方案，这些方案在没有CFL条件的情况下保留了质量守恒性和正性。对于非线性方案，使用Brouwer的不动点理论证明了适定性，并且我们展示了Picard迭代的收敛性。我们还研究了两个离散的Lyapunov泛函，平衡点的渐近稳定性和局部稳定性。此外，我们将离散半群理论应用于误差分析，并以\（L ^ p \）范数求出收敛速度\（O（\ tau + h）\）。理论结果通过数值实验得到证实。