Non-Standard Discretization of the Advection-Diffusion-Reaction Equation with Logistic Growth Reaction

The goal of this work is to make a comparative analysis between the standard finite difference method and the non-standard finite difference method, then to make a non-standard discretization of the advection-diffusion-reaction equation with a reaction modelling a logistic growth which can be the evolution of the concentration of a microbial population in a medium, the equation will thus model transport and diffusion of this population in the aforementioned medium in one dimension of space and one makes numerical simulations to compare the non-standard scheme and the Euler’s scheme, explicit in time, implicit for the first order derivative in  and centered for the second order derivative in . One arrives by constructing a scheme of the advection-reaction equation, then adds the term of diffusion to obtain the non-standard scheme