In this paper, by introducing two temporal-derivative-dependent auxiliary variables, a linearized and decoupled fourth-order compact finite difference method is developed and analyzed for the nonlinear coupled bacterial systems. The temporal-spatial error splitting technique and discrete energy method are employed to prove the unconditional stability and convergence of the method in discrete maximum norm. Furthermore, to improve the computational efficiency, an alternating direction implicit (ADI) compact difference algorithm is proposed, and the unconditional stability and optimal-order maximum-norm error estimate for the ADI scheme are also strictly established. Finally, several numerical experiments are conducted to validate the theoretical convergence and to simulate the phenomena of bacterial extinction as well as the formation of endemic diseases.

翻译：本文通过引入两个依赖时间导数的辅助变量，针对非线性耦合细菌系统，提出并分析了一种线性化、解耦的四阶紧致有限差分方法。采用时空误差分裂技术与离散能量方法，证明了该方法在离散最大范数下的无条件稳定性与收敛性。此外，为提升计算效率，进一步提出了一种交替方向隐式（ADI）紧致差分算法，并严格建立了ADI格式的无条件稳定性和最优阶最大范数误差估计。最后，通过多个数值实验验证了理论收敛性，并模拟了细菌灭绝现象及地方病的形成过程。