Qualitatively stable finite difference schemes for advection–reaction equations

A systematic procedure is proposed and implemented for the design of nonstandard finite difference methods as reliable numerical simulations that preserve significant properties inherent to the solutions of advection–reaction equations. In the case of hyperbolic fixed-points, a renormalization of the denominators of the discrete derivatives is performed for the numerical solutions to display the linear stability properties of the exact solutions. Non-hyperbolic fixed-points are described with the help of two new monotonic properties the construction of schemes, which preserve these properties, being done by nonlocal approximation of nonlinear terms in the reaction terms.