Maximum-Norm Error Estimates of Fourth-Order Compact and ADI Compact Finite Difference Methods for Nonlinear Coupled Bacterial Systems

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. In particular, an adaptive time-stepping algorithm is developed and tested for long-term stable simulations.