An ADI method for hysteretic reaction-diffusion systems

In this paper we consider a mathematical model motivated by patterned growth of bacterial cells. The model is a system of differential equations that consists of two subsystems. One is a system of ordinary differential equations and the other is a reaction-diffusion system. An alternating-direction implicit (ADI) method is derived for numerically solving the system. The ADI method given here is different from the usual ADI schemes for parabolic equations due to the special treatment of nonlinear reaction terms in the system. Stability and convergence of the ADI method are proved. We apply these results to the numerical solution of a problem in microbiology.